The rear is an open diff as well on MK7.Heck even Toyota intentionally made the rear axle 0.7% taller on the GR Corolla and uses the center diff clutch pack to make it all work on the road with the idea being that it will more easily rear bias at hard throttle for handling.
Honestly with VW I'd be more worried about the tire pressure system getting mad just because it counts tire rotation, though setting pressures and then resetting the system from the radio menu should more than account for that. Slick surfaces would trigger the tire pressure warning for me in the GTI when it would have random slips I didn't detect and it would see consistent small wheel spin at low speeds.
The side to side difference would be best to deal with on the front since the AWD VWs retain an open diff on the front and I believe that the rear uses a clutch pack to engage it. I'm more familiar with the new one in the MK8 R so I'm not too well versed on the rear setup on the earlier cars, but they've all retained the open front and use the brakes to simulate limited slip.
Yes, you reset the TPMS anytime you change wheels/tires or have a pressure correction. I've never seen it kick on otherwise unless it was a low tire....even on track when you are all-four wheels a-blazzin'!