In most cases, it should; the damage is on the outer glass portion only. The touch portion may not be entirely functional, not to mention it could easily cut your finger with the glass.
Correct, Apple will not help you. FYI, though, the iPhone's screen is glass, not plastic. It was originally plastic when it was shown at Macworld 2007, but then Apple announced they had changed it to a glass screen.
The display assembly on the iPhone (the 2G version, which from your photo I can see you have) also contains the LCD and the digitizer, so a) it's going to cost over $100 and b) you can't buy JUST the glass. Apple realized this when they developed the 3G and made the glass separate from the rest of the display.
If you're interested in doing it yourself, I'd check iFixit or another one of those parts places and see if they've got it (it'll be sold under "display assembly"). I don't recommend doing it yourself though. I tried doing that with my first iPhone and ended up buying a new iPhone. Apple may do this for you, but they'll charge you somewhere around $250 if I recall correctly. There are some other companies that'll do it for less, but they're not authorized by Apple; seeing as the warranty on most 2Gs is up anyways, I'd look for one of those.
Good luck!
won't replace it but you can always get a screen replacement kit off Ebay and try it.. or send it in to one of the places listed since your warranty is void now anyway..