I have serious reservations about omni antennas and almost always recommend against them. I have known sat ops at the top of the leader boards that used a small yagi type antenna handheld or on a tripod sometimes from inside their home that performed better than omni antennas. That said, the problem is almost always receive. I used to have a joke... how do you know a sat op using omni antennas is on a pass? You hear a lot of "QRZ? Is someone calling me?!!"

In all seriousness, I would be reluctant to even try it without preamps and I just hope that you are satisfied with the results after going to all that expense and effort.
As to your questions,
1) Specific recommended mast-mount UHF LNA with T/R switching (max RF output of IC-9700 is 75W)? I realize that best practice is to use much less than full output.Most mast mount preamps I know of have been discontinued and what's left is likely very expensive. I used the ARR RF switched regular preamps inside a box I purchased and mounted to the mast. They are still available ($149.95 new, ~$100 used, if you can find them) and I coupled them with some MFJ-916Bs for filtering out third harmonic issues and out of bad overload. Catch is, 25 watt output limit. If you e-mail me, I can send you pictures.
2) Should I also consider a separate mast-mount VHF LNA with T/R switching (max RF output of IC-9700 is 100W)? As above, I realize that best practice is to use much less than full output.With an omni antenna, I would... you will likely need help hearing even the VHF downlinks.
3) Pros and cons regarding powering the LNA(s) from the IC-9700 versus feeding DC from a separate bias-T arrangement? Can the 9700 power both UHF and VHF LNAs if I decide to get both?
I ran a length of three wire rotor cable hooked to a 12v supply with a switch to choose between the preamps... black was common ground.
4) Based on your experience, are there any other word of wisdom or “gotchas” I should be aware of?Just not to expect too much from omni antennas. If I were you, before I spent all that money on preamps, feedline and expensive omni antennas, I would get a simple handheld antenna (buy or build) and just get on the birds and learn about them first. For that matter, you might also build some simple omni antennas from PVC and copper wire and compare them to a handheld yagi so you can see what the difference is like.
5) FYI: My QTH is mid-latitude (FN20), and my proposed feedlines are 75-ft of LMR400.Probably good enough on the feedline, preamps will easily overcome that 1 & 2 dB loss. - What to tell you to expect from your QTH given your antenna choice, not sure. Normally, guys in the North East do great into Europe, but that means low horizon passes and I have no idea if those antennas can make those passes. I see your issue is with the XYL... (Sorry, I have no experience with that, my XYL couldn't care less and even got her license, N4ICY.) Usually the issue for guys is a lack of AZ-EL rotors and we convince them to use a cheap TV rotor and set a yagi at 15-20 degrees tilt. That usually works a whole lot better than an omni setup. In fact, I started out that way, with simple homebrew directional antennas and no preamps, and worked into EU and even Africa over the SSB birds at the time... but forget the one FM sat back then (SO-50), couldn't hear it's downlink without a fully aimed antenna.
Please take all this in the spirit it is offered. Don't want to discourage you, but also don't want you to have a hole in your pocket and then be doubly discouraged. What I would really like is for you to prove me wrong and get me to put up a set of those antennas here! But even if you did, I doubt that would happen. After selling off my AZ-EL setup that helped my earn the AMSAT Century and Gridmaster awards, I find myself cross country with a very different QTH... surrounded by trees (leaves are a big deal on UHF), a steep hill immediately behind my house and a horizon full of mountains. I tend to be more interested in portable ops these days. If I can find a spot to go without same.

Hope this helps... 73, Kevin N4UFO