the 225 model has quattro and a slightly more robust 1.8 with a slightly bigger turbo (borg warner K04 vs K03 in the 180), the 180 is FWD. Quattro is preferable, though it's a Haldex based system, which means it's FWD until it needs more traction. There are modifications that can be done to force it at the ratio you want up to 50/50 (haldex controller)
FWIW it's basically a golf chassis so there are a lot of interchangeable parts.
Understeer is a persistent problem, will take a lot of testing / tuning to get it where you want it, it's not an easy task. The usual strategies apply.
I have been out of Audis for a long time now, so not really in touch with what people are up to today, and what's for sale. Mk 1 TT are not really a hot item so you should be able to pick one up for less than a broken miata that are priced ridiculously IMO (I'm just salty I can't buy a POS first gen miata for reasonable money)
Hit up the forums...
Local Austin group TexasAudiGroup (looks like they're facebook only now https://www.facebook.com/groups/394300721981/ )
I like the old school 1.8T, it was before direct injection and is stoutly built (no issues with horrible carbon build up that has to chiseled off or blasted with walnut shells like the newer Audis). Oil coking can happen of course as the little bastard turbos run pretty hot, clean it up and get rid of the shit EGR, add a catch can etc. Not too hard to find a good shop (or DIY) to stroke it out to 2.0, fuel it properly and let her eat a good sized turbo if you're so inclined. Otherwise if she's in good running condition, just a "chip" to crank up the boost wakes them up decently. Heat soak is annoying as with most cars in the summer, especially boosted cars. The usual upgrades apply to combat that.
Weird VW/Audi gremlins occur especially at this age of vehicle, and CEL can be a near constant companion. Just refresh everything ...
Timing belt needs to changed every 60k or so (book says 90k, don't listen to it). Interference top end so if the belt snaps, things bend and break unpleasantly