Rich,
You ask some good questions, and I hope to provide you with some good answers....and, I think I'll also give you even more info than you asked for, if that's okay? (I promise to try to NOT add more to spin your head even more!)
Please allow an honest (if somewhat sarcastic) preface:
--- You can have easy, efficient, and cheap....as long as you only pick two of these!
(meaning, you can't have all three in the same system, no matter what the marketing-hype might imply.....so be prepared to compromise somewhere...)--- And, please, please don't get bogged-down in the math / engineering in this! Yes, there is some, but for the most part, if you follow at least some of my recommendations, you can gloss over most of the math/engineering here (at least to start with)....this isn't rocket science, and you do NOT need to spend hours delving into the minutia, yet.
--- Further, unless you have absolutely no shading of any part of any panel, wire them in parallel not in series....and, eat the cost of larger wiring....(see below for details)
1) First, you must do an energy survey / write-up an "energy budget".....figure out how much energy you need, over a specific period of time, and how long will you need to provide this energy....(even if all you want is to run your frig, do an energy budget, so you will know what you need)
Without this, you're just spinnin' your wheels.

2) A few
specific answers / clarifications here:
I currently have four 100W 12 volt Siemens solar panels. They each put out about 17VDC.
<snip>
I'm thinking of putting together a small system for emergencies using all four panels. Pack it all away in the basement for a rainy day (EMP). I've read articles until my head is spinning so I thought I start a conversation and soak in the ham knowledge here.
As usual, cost is everything and I don't intend to run a welding shop or the whole house on this system. Anything that will support some wall wart chargers for various electronics, maybe a laptop (which I never use except on AC because the battery isn't great). Running a fridge for a while during the day would be an extreme high-end load for this
All the above is understood, and is all very do-able!
And, it's pretty inexpensive to do....(mainly 'cuz, as long as you don't need to run high-power/high-current devices, you can run some/many things directly off 12vdc batteries, use small/inexpensive DC-AC inverter, and use smaller/less-expensive storage batteries)
or maybe the microwave in the camper.
Hmm....still do-able...
But, due to its high-power/high-current use, it will require a large inverter and larger storage batteries .... (see details below)
Some questions:
Is it even possible to run this 400W solar array directly to an inverter and skip batteries altogether, understanding it will only work in full sun?
Nope, that isn't going to work...(although, in an emergency, you could give it a try....it's possible, but unlikely, that you could force the inverter to work for a short while....but, in the unlikely event that you could get it to work for a few minutes, chances are it would be damaged afterward....so, the answer is a qualified No.)
See below, for details on understanding that this IS a "system", no matter even if you just want it for use in times of emergency (although not on a "rainy-day", as solar doesn't work as spec'd under overcast skies!)....but, 'til you read the details below, please understand that, if you must run some 120vac appliances (such as your home refrigerator) versus running things off of a nominal 12vdc, then you must have some batteries to run an inverter....
And, fyi...you "size" the inverter based on a few major factors:
a) the highest load any device you will need to use (frig, microwave, etc.) could draw...
b) the real-world capacity of your batteries (and/or solar array) can provide to the inverter, over a specific period of time.
If I have to use batteries, can I use a couple of plain old SLA 12v batteries, maybe 100 Ah each, just because they're cheap(er)? You "could" use any nominal 12vdc battery....but, please understand that "plain old SLA 12v batteries" generally are not designed for extended deep-cycling, and while they'd work for a few cycles in a pinch, for more regular use they'd fail quickly (and will end up costing you many more $$$ than you think)....and, if this system is supposed to "work when you need it to work", then they'd be a bad choice.
Simple fact is this....for least expensive, and very-good capacity, still to this day nothing beats "golf-cart batteries"....yes, a pair (or two-pair) of 6 volt GC2-sized golf-cart batteries are still almost unbeatable in the application of daily-use / daily-charge (or almost daily use 'n recharge)...a typical pair of 6volt golf-cart batteries will provide you ~ 220 - 245 A/H's of capacity for ~ $190, all-in....and, if charged / kept charged and/or used-n-charged daily, they will last you years! (fyi, a local battery shop here will deliver to you a pair of Crown 6v GC-2 golf-cart batteries for $95 each, $190 for the pair, delivered!) But....
But, of course, these cannot be left alone (not being charged) for months, etc....as they do need to be charged / kept charged, and maintained (water added) regularly....so, some folks start looking for an "easier" solution, such as "sealed" battery....and, yes, I also have experience with them (sealed AGM batteries)....
A sealed AGM has a much lower "self-discharge" rate, but they do still need to be charged regularly (or better, charged all the time) and make sure they are never allowed to be in partial state-of-charge for more than a day, otherwise their capacity will decline quickly (yes, this is true....despite what the manufacturers may try to tell you when marketing them....have a look at the real spec sheets, manuf recommendations, and their warranties....and, you'll see...they do work, but are more critical to being kept at 100% state-of-charge than simple "flooded" batteries, like golf-cart batteries)
However, in your specific application, if you can get them to 100% charge (and keep them charged) often (daily?), these would be a good choice for you....certainly more expensive than golf-cart batteries, but low-maintenance might trump the cost? (that's a decision you must make)
Sure, Lithium-Iron-Phosphate (or some other lithium-compound) batteries. size-for-size and weight-for-weight, have higher capacities and can be discharged to much lower state-of-charge, as well as maintain their nominal voltages for greater time period while being used, and can be left partially-charged without serious effect....but, they are pricey and require battery-management-systems (BMS) to maintain perfect cell balance and perfect individual cell-charge, etc...(and while some of these BMS's are included with the batteries, almost all BMS's seem to produce some RFI, many of them produce a great deal of RFI!)
To be clear, most of these newer Lithium-Iron-Phosphate batteries have VERY low self-discharge rate, and can be left alone for months without charging as well as left in partial state-of-charge, without ruining them!
But, as I wrote, they're pricey! (and need BMS)
So, while they would be a great choice for your application, the ain't cheap!
{are you starting to grasp my preface above?}
How many batteries would make the most use of the 400 watts available or is that a function of the inverter output? Actually, you need to start your design with the knowledge that this is a "system", not just some various devices wired-together....with sources (solar array, batteries, commercial AC Mains, generator, wind power, etc.), storage (batteries), and MOST IMPORTANTLY the energy "consumers" (such as refrigeration, radios, lights, etc.)....
And, you generally size the "system" according to your energy needs (how much you consume, and over what period of time) AND how much room/space you have for unshaded solar panels (and room/space for batteries)....and, of course what is your budget!!
I have this bundle in my Wish list:
https://smile.amazon.com/Renogy-Converter-Controller-Parameter-Adjustable/dp/B09QKM28H9/ref=sr_1_10?crid=C7V6KSW8P2GY&keywords=mppt%2Bcharge%2Bcontroller%2Bmppt&qid=1646100641&sprefix=morningstar%2Bmppt%2Bcharge%2Bcontroller%2Bmppt%2Caps%2C337&sr=8-10&th=1
I'm not a fan of Renogy....poor QC, usually lots of RFI, and pretty cheaply-made-overall...in my opinion, not reliable enough to rely on for emergency situations...
Couple of questions on that:
Is 2000 watts just a ridiculous expectation for my 400 watts of solar?
In general, yes....
Or does that just mean I need more batteries to make the most of the 2000W? i.e., fewer batteries would discharge quicker with that kind of load?
In general, yes....but, you can run a 2000 watt inverter for very short periods of time (a couple minutes) with smaller battery banks of ~ 250 A/H (see some details, below), but not very advisable!
If I wire the panels as a 24V array, 2 in series and 2 in series and then the two pairs in parallel, do I use that current rating as the input for the charge controller?
Well, yes...assuming your series-connected max voltage is within spec of the controller voltage rating...BUT.
But, unless you absolutely have NO shading of the panels, series-connection is a bad idea!
Yes, I'm aware this is a minority view....but, please understand that the majority of folks' experiences here are in fact with rooftop / barn-roof / out-in-the-clear solar panel installations, where shading is all but nonexistent, so they have no experience with panel shading and/or the negative effect is either not noticed by the end-user (or even worse, the so-called "experts" designing or selling these systems, simply don't care about the end-user's satisfaction, and all they care about is saving $$$ on using smaller gauge wiring)....please see details below (some diagrams directly from the panel manufacurers), before you flame me...
It says "40 AMP" so is that the max it could draw from the panels?
Good question...ask Renogy...
But, in general, no....this is usually the max output spec (max current into the batteries)....
But, since I'm not Renogy, I cannot give you a 100% certain answer about their marketing specs....so, again, ask them....or better idea, buy better controllers!
I know they won't put out that much but is that a specification that's meant to stay under or to make sure I can deliver?
Not sure what you're asking here....but, in general, you size the controller to your panels / array size....I think that answers your question?
Wiring as 24V means I can get by with smaller wire, between and from the panels to the charge controller?
Yes....but, as I write.....unless you have absolutely no shading of any part of any panel, do not use series-connections, just wire them in parallel (and pay for good/proper-sized wiring)....
10ga? 4?
Wire size depends on length (round-trip length) and current that is being carried....so, that's an answer you need to look up on a chart, based on your wiring length and your max current expected (figure 35 amps max...even if you'll only ever see 30 amps out of four 100-watt panels)
Thanks in advance. 73
3) Next, right after doing your energy budget, please understand that solar panels only produce usable power when they are not shaded! So, place the panels out in the clear, where they are NOT shaded at all during the peak ~ 5 - 6 hours of the day (local noon +/- 2 to 3 hours), during the entire year....and, if possible, not shaded at all, at anytime of the day, the whole year long!
(fyi, Montana has shorter winter days and longer summer days, than we do down here....and, much lower sun angles....so, judge array size and location accordingly....btw, while you do have fewer hours of sunshine in winter up there, and lower sun angles, too....understand that the panels work great in cold weather! Cold air temps and clear-skies, especially high-altitude locales, are GREAT for panel output!! So....so, while your sun angles are slight, your air temps are wonderful!)Trees, etc. shading the panels, will kill your solar output! And, please note, it doesn't take shading the whole panel to kill its output!!! Shading just a small part of a panel, will do it!

Have a look here....Here is an image (courtesy of Kyocera Solar) showing the shading needed to reduce panel output by 50%:

And, here is an image (courtesy of Kyocera Solar) showing the shading needed to reduce panel output by 100%....yep, zero output, or darn close to zero!

And, here is a short video (done by some friends of mine, who make MUCH better videos than I do...I'm much better at the engineering, they're better at videography!) which show the above shading in real time, as well as the serious issues / effect of running the panels in series (which is why I recommend parallel, unless you can guarantee they'll never be shaded!)....sure, while running the panels in series does save some $$ on wiring, and can save a bit in voltage drop, in series-wired panels if you have any shading at all, your output is all but zero....so, paralleling panels is almost always best (unless they are all mounted up and in the clear, without any shading)...
https://www.youtube.com/watch?v=1qD3mN8VotQSpend the 12 minutes, and watch this video!
So, after you've made-up your energy budget / consumption budget, before you spend anymore time running the math....nor any money at all, nor any more time "researching" this....go outside and find out where you're going put these panels....how are going to keep the shade off of them....(and, keep the snow off of them in the winter, etc.)....how are going to run the wiring....where are you going to put the batteries, charge controllers, and where are going to put the inverter, etc...
You really need to do all of this first....'cuz, until you have all of that figured out, the rest of this is moot!

Then, go and look at your energy survey / energy budget (you can use the power consumption / current draw specs on your devices like refrigerator, etc. and estimate how many hours-per-day that each of those devices will need to operate)....and, then you can start to figure out how much money/time/effort you wish to invest in this.
(I suspect you'll go-for-it, using a much smaller inverter, and even some native 12vdc lighting/chargers/devices....along with a couple golf-cart batteries, and you'll be good-to-go!)
In my life on-board, where everything (except the seldom-used air conditioning and domestic hot-water) is 12vdc (and I cook with propane), we just boil everything down to "amp-hours-per-day" at 12-13vdc.....(abbreviated A/H-per-day, but some just take the short cut and say "amps/day") But, on-shore, many use "watt-hours-per-day", just be 100% sure you calculate your watts correctly, and understand the losses / efficiencies of any inverters...
4) BTW, just about everyone (including me) now-a-days uses MPPT solar charge controllers....which keep the solar panel outputting at its peak power output while providing the batteries with their maximum possible charge current....this is all good....just remember that the "cheap" ones should be avoided, as they usually produce serious RFI as well as not being as efficient....(I use Blue Sky 2512ix controllers.....but some of the Morningstar MPPT controllers are also good)
5) Rich, please know there is a lot to all of this....but, the main problem for most laypersons is that there is so much BS out there....and, unfortunately some comes from supposed "experts"....and, worse, "internet experts" who may be well-intentioned, but have never done any of this in the real-world...
So, before you spend the time / effort and $$$$ to move forward, have a look at what really works, in the real world....and, that does not include the "local guys selling solar for your house", 'cuz 99% of them are salesmen that have no clue about solar systems!

6) I hope you don't mind me adding some of my personal experience here....just to show you that there ARE some of us out here that have done this....and, we are willing to share our experiences / recommendations....especially to my fellow hams and sailors....(yes, I know the math / engineering behind all of this....been doing it for decades....but, I also walk-the-walk....meaning I have lived this, in the real world, for decades, not just "a guy who bought some panels"...hi hi...)
While I played a bit with solar as a teenager in the 1970's (and installed a panel here and there, usually on boats....but they were wickedly expensive!!!)....I designed/installed my first professional remote/off-grid solar array in 1984, and while I did a few more land-based systems over the years, most of my solar array designs/installs in the past 4 decades have been on-board offshore sailboats (i.e. those that sail across oceans and stay in remote locales for months, etc.).....and, these days (last 10 - 15 years) it has become fairly easy to generate (and store in batteries) sufficient power to run all basic electrical systems/devices....
On my present boat, my panels / array are now nearing 15 years old....they've survived a few crossings of the Atlantic ocean, a few extreme storms / full-Gales, one serious tropical Storm, and dozens of severe summer T-Storms....out in the hot/sunny/wet/salty environ of the ocean being only a couple feet away!
And, as I wrote....in my life on-board, where everything (except the seldom-used air conditioning and domestic hot-water) is 12vdc (and I cook with propane), we just boil everything down to "amp-hours-per-day" at 12-13vdc....and, as long as the sun shines at least every 3 days or so, my present 520 watts (and ~ 900 A/H of batteries) provides all my on-board power for refrigeration and freezer, all electronics (including navigation, radios, radar, etc.), lighting, fans, etc....(I cook with propane)....this is not camping-out, this is luxury-living on a 47' sailboat (think of this like a 2-bedroom/2-bathroom house, with large living room/kitchen, huge patio, etc....lots of fresh water, cold beverages and hot food....and, I can go anywhere in the world)
While I do plan on upgrading to ~ 750 watts of SunPower panels (and ~ 250 watts of old-school / Kyocera panels), with all new controllers....my current set-up is 520 watts (four 130 watt Kyocera panels), all up in the clear, unshaded....with two Blue Sky 2512ix controllers....all working well.
Have a look:






I do hope this helps some?
73,
John, KA4WJA