Do I need to open the games in Galaxy in order to get achievements, or do the desktop shortcuts works as well?
You'll need to open the games in Galaxy.
The desktop shortcut that Galaxy creates should -as far as I've understood (meaning, I can most certainly be wrong)- open the game through Galaxy (or at least in a manner such that Galaxy is aware that the game is running). However, if you installed the game from its stand-alone installer, then, yes, you probably need to open it through Galaxy as the shortcut most likely points directly at the game.
The desktop shortcuts created by Galaxy, don't point to Galaxy. Same case with the start menu shortcuts.