Unreal Tournament 2004 and Quake Live are installed through my PC, but not through Steam. How can I sync achievements with my Steam profile for these and other games which Steam sells but which I bought through another vendor (or in Quake Live, it is free and browser-based)?
The "Steam Version" of games differs from the non-Steam version - if your game wasn't purchased or activated through Steam, you can't get Steam achievements for it. I don't believe there's any way to (legitimately) take achievements from a non-Steam version of a game and import them into Steam.
Additionally, Quake Live and UT 2004 don't have Steam achievements, as far as I can tell. In Quake Live's case, they have a completely separate achievement system (Awards on your Profile) and UT2k4 was released prior to the introduction of the Steam achievement system, and I don't think it was ever modified to include achievements.