The fastest thing to do is find a knowledgeable local ham to help you.  It could be any number of things.  Good luck!

I always get good audio reports with a Heil HC-4 (HC-5 better for ragchewing) on a TS-930S and a TS-700S.  I've only used a TS-830S on CW, but I'm sure they would work well with it also.

Good grief!  Don't pay anybody to do the paperwork for you.  There are instructions on the web:

http://www.arrl.org/applying-for-a-vanity-call

The price is always changing, but it's presently $15 for the 10-year term of your license grant paid at the time of application and renewal.  Be sure to pick a valid and available callsign, though.  If you want a good laugh, take a look at the rejected applications sometime.

This year was my first visit to Timonium and I have to say that I was underwhelmed at the time, but in retrospect, it was a decent hamfest.  I'm only about about 45 minutes away and that was totally worthwhile and a good outing with my toddler son.  There was not a lot of decent used ham gear, but there was a lot of stuff that would interest the tinkerers.

On the surface, this seems like a fair comparison, but these are two fundamentally different radios in completely different operating classes.  If you like one, you probably won't like the other.  The K3 is engineered for raw performance under very specific (crowded) conditions in a portable package.  The fit and finish is not the same as the JA radios (although I really like the ergonomics of my K3).  It's smaller and lighter than the FTDX-3000 in addition to the difference in performance numbers.  For me, the K3 was a no-brainer even versus the very similar and very competitively-priced TS-590S.  I would never use the bells and whistles ("features" you called them) of the FTDX-3000. 

I buy from The RF Connection.  They only deal in connectors and cable and they know their stuff.  Even if it's not on their web site, they can get it.  Cheap CB-quality no problem.  Spacecraft-quality no problem.

I used one of these for a while (hard to argue with free) to turn a 12-element K1FO 2-meter Yagi and a 3-element 6-meter Yagi.  No trouble at all, except you had to manually peak on stations because the box only reads out in 45-degree increments.  It doesn't have a brake, which is problematic for really long antennas.  Like the others say, the controller is not compatible with the newer bell rotors.  

Good suggestions mostly.  The best suggestion is to make it so your equipment is accessible from behind.  Because space is at a premium, I have lawn furniture stored behind my desk.  But, it's easy to move out of the way for major wiring tasks.  K1IR has photos of his operating desks which are on wheels so he can do wiring behind them.

I would avoid elbows in your conduits if you want to use anything other than flexible coax.  One way to get around this is to have a large outdoor junction box where you bring the (semi-)rigid cable up for grounding and lightning arrestors, as well as the conversion to small flexible cable.  Properly done, this also provides convenient and expansion-friendly weatherproofing as well.  KA9FOX has photos of his setup for a good example of how to do this.

Another thing you might consider is having an "amp room."  This keeps the roar of blowers out of your shack.  But, that depends on how serious you are!

This is an academic discussion at this point.  What are you trying to do on the air?  But, without any further guidance, old "contest-grade" radios with appropriate filters are probably your best bets, some examples:  TS-830S, TS-930S, IC-765 (might be a bit high in price), FT-990 (also probably a bit high).  Most of these rigs will also take an aftermarket Inrad roofing filter, something that will definitely improve performance (speaking from experience with my TS-930S).

The Sherwood list is great, but it's not perfect.  He has some caveats on the site (about the small sample size, etc).  But, the bigger thing is that his tests only approximate a single real-world situation (trying to copy a CW signal at the noise floor N kHz away from a very loud signal).  That may not be at all relevant to the operating you wish to do.

Buy the biggest power supply you can afford, lift, or tolerate in your shack.

Performance.  All of these smaller all-in-one radios have credible/useful receivers, but they will get squashed on a band full of very big signals.  As long as you're using simple antennas (mobile whips or wires), you'll only run into this occasionally.  By "squashed," I mean that the sensitivity to weak signals will be reduced in the presence of other louder signals nearby or you may hear signals that aren't actually there.

I agree that you should look at the used predecessor to the Icom IC-7000, the Icom IC-706.  There are three versions, IC-706, IC-706MkII, and IC-706MkIIG.  It's arguably one of the most popular ham rigs ever made so there are lots of them out there.  Try to find the MkIIG.  I'll let someone else explain the differences between the three.  The Yaesu FT-100D is a bit long in the tooth and has some reliability problems with the amplifier section, but is probably the best receiver of all of the radios in this class.

For power supplies, Astron is a good recommendation.  You should be able to find a used Astron RS-35 for less than $100.  Buy local because they're heavy!  I had one for many years.  I use the Astron SS-30 in my home station and the MFJ-4125 for travel.

Welcome to ham radio! 

The answer depends a lot on your local ham culture.  If the VHF crowd is cliquish, you won't have any fun with a handheld or a mobile.  I like to just have the chatter on in the car because often the guys on the repeater are more insightful than the people on the broadcast radio.

HF from an apartment could best be described as something between excruciating and tolerable, depending on your situation.  I lived in an apartment for two years and only operated HF mobile and from a club station.  RFI, both giving and receiving, is a fact of life in that situation.

If you got into ham radio to contact far-away places by radio, than skip the VHF stuff.  The happy medium would be to get one of the all-in-one mobile radios like the FT-857D or the IC-7000 (or their predecessors).  Given your situation, you won't notice their shortcomings and they'll give you maximum flexibility to explore the bands.  If you have more questions, feel free to ask.

Not only does the 8-ft fiberglass arrive fast, the shipping is practically free.

I have a tricked-out (four Inrad IF filters, Inrad roofing filter, PIEXX digital board) TS-930S, which is considered by many to be better than the 940, although they are quite similar.  The 930 is a great radio.  It sounds like a million bucks on both TX and RX, has a pretty stout receiver, etc, etc.  I bought a K3 this past summer and although I still love the 930, the K3 kicks it all over the place as far as receive performance on a crowded band, even with the stock 5-pole 2.7-kHz roofing filter and my modest wire antennas in the suburbs.  

The point is that buying the right modern rig will give you considerable improvement over the TS-940S. I've never used an 870, so I can't comment on that.  But, I think it's fair to say that any of the new rigs in the $1500-$2000 range (TS-590S, TT Eagle, K3) are going to be head-and-shoulders above your 940.

I second the suggestion of the TT Eagle if the K3 is out of your price range.  I know a couple of guys who have Eagles and they love them.  I was not aware that it has an IF-out jack now.