Say I play a game on the Xbox 360 and earn some achievements. If I later buy the same game through Games for Windows Live (under the same gamertag), will those achievements show up in both places, or are they platform-specific?
As long as you sign in with the same Live account, your achievements will be visible on both sides of the fence. That is, when playing a Games for Windows Live game, you will be able to view all achievements in all games for both platforms.
However: The Xbox 360 and GfW-Live versions count as separate games, even though they will have the same title. So, if you play Fallout 3 on both the PC and the 360, they will not share achievements. Yes, this also means that you can earn a total of 2000 gamerscore by earning all of the achievements on both platforms.