As I mentioned earlier, in order to get correct results form Groove clip looping, you need to start by making sure the clip starts and ends on or very near a beat, and if it changes length by more than a beat, you need to check the Beats in Clip value in the Groove Clip section of the Clips tab in the Track Inspector, and set it to the correct value.
Enabling Clip Follows Project in the Audiosnap Palette should not immediately change the length of the clip unless the mode is something other than Autostretch, even if the tempo is not matched to the project, and no matter where the clips start and end or how long they are.
At this point, it might be helpful to know what's in the project, and how it was created. Is this all audio that you recorded into the project to a click? Or was it imported with or without matching the existing project tempo?
Regarding the snap problems, make sure the To/By switch in the Global snap module is set to To, and disable snap to Landmarks and Zerocrossings.