Yes, you can autox an R888R or NT01 but it'd be a bit odd. They would likely do quite well compared to your average tire but I don't think they'd do as well as the RE71R or BFG Rival-S 1.5. The reason it's odd is that they're not competitive tires, the treadwear rating is too low for street class but they're not as sticky as full slicks so they fall in a no-mans land and not many people use them.
The reputation of the RE71R is if you can drive near the limit for a whole lap on track it will overheat and get slidey and wear out prematurely. Track guys basically treat it like a qualifying tire for time trials. If it's your first track day or you don't have a bunch of experience you likely won't get anywhere near driving the car that hard or fast.
I think you're on the money with your last thought, buy a track capable tire and run it everywhere, yes it won't be as grippy autox-ing but it will last and give you experience and seat time which is far more valuable when starting out than experiencing ultimate grip. Also that means you can drive those tires on the street so less time wasted changing wheels before/after events.
Again if you're just starting out, I would totally advise against running the 17x9, low grip teaches you things at lower stakes and you'll have a better understanding of car control before stepping up the grip levels. Stock wheels and tires hold up to a lot of abuse and people dump their OE tires for cheap if you managed to wear yours out or if someone put crap tires on the car you bought used.
But hey, you do you, I'd suggest a ~300TW track-capable tire to start out with, Michelin Pilot Super Sports and Continental Extremecontact Sport are the leaders on that front but to save a few bucks you could do Firestone Indy 500 or Sumitomo HTR ZIII or some other tire I've forgotten like a Bridgestone Potenza something (RE-11 is still available but that's an old outdated tire).
Stickier tires would be BFG Rival (non-S), Hankook RS4 (or RS3 if some are still available), Dunlop Direzza ZIISS, Maxxis Victra VR1, etc. etc.
This is good info too:
http://www.ft86club.com/forums/showthread.php?t=25485
Go to Tirerack and put in tire size 245-40-17 and take a look at your summer tire options and you can back check here, although I think I mentioned all the common trackable tires people usually use...