The gauges are pretty much only there for reference, not to be treated as accurate. As long as the gauges are showing the same results under the same operating conditions, don't worry about them. When they start showing you something different, then you start worrying.
Feltz tire is 100% legit and a pleasure to do business with. I bought some tires a few years back and the owner called to arrange shipping to me (to a loading dock an hour or so away, but that's not terrible around here)
The tires and run flats are 2 different components of the same system. The only thing remotely special about the tires is being treated to resist ozone damage and being labeled as such (you should see a big "OZ" on the sidewall)
20 year old MT tires aren't terrible. Most of it is how the tires were stored and whatnot.
I have 20+ year old MT tires on several trailers and don't worry about them. Just make sure they are sufficiently aired up and you should be okay. J
Since yours are on the truck itself I'd just keep an eye on them until the new tires come in, and then I would keep the best 1 or 2 as emergency spares if you didn't buy a spare from Feltz.
As for your pump rebuild, did you check reviews on that place first? A few people have posted decent experiences, but many have also given poor reviews for excessive turn around time and whatnot.
The usual recommendation is to buy a new one from a legit vendor, such as Hummer Parts Guy (new pump on sale just under $1700 right now)