Is there a way to see how far you've progressed on the various do x thing y times achievements (like the ultra / super combo finishes, flashy background wins, etc)?  Also, can anyone tell me if those can be done anywhere (training, challenge mode, arcade, online) or only in certain modes (online only, etc)?

Hmm, it doesn't really specify actually. I suppose you can pretty much do them all in training.

Training sessions don't count. You have do do them in either arcade mode or the on-line modes.

Here are a couple of sights to help ya, they have been invaluable to me:

EventHubs

MyCheats

Street Fighter Dojo : THIS sight is great. It has video of a lot of top players around the world playing each other with different characters. So when you read about the strategies you can actually see them in action. The first two I've listed plus this one have helped me out a lot, and I've been playing SF since forever. If you want to send me a friend invite please do so. I play SFIV all the time and I'm always up for casual matches. My tag is EnderzGame-just like here. =) There is also a great community on the XBox 360 SFIV board as well if you're looking for more players.