Derailleur hanger moves when tightening quick release
Edited from my comments above:

The first thing I would check is whether the hanger is removable.  

If it is removable, I would check that it is still bolted securely in place. It's pretty common that removable hangers sort of overlap with the dropout. So it's possible that the hanger is loose, and the QR is then clamping it firmly in to place, but then each time you remove and reattach the wheel it clamps in a slightly different position.  That could explain the tricky shifting.