As Munkeh said, if you move along the XMB, it should be between the "Game" section and the section that olds the Store. In whichever section that is called (I've forgotten), there should be an option called "Downloads" or something. If you go into that, you can see the download status.
If you turn off the PS3, it will save whatever you've downloaded. If you play other games, it will somtimes keep downloading and sometimes won't. In most instances, it will stop downloading if you start playing your game online. Some games, however, stop the download even if you're not playing online.