Normally yes, unless a specific transmission mode has been fixed
But the tester also shows that. Crossed pairs are nowadays also automatically corrected (no need for crossover cables anymore ;-))
Yes, that should be the case. Although I wondered what happens if only a CAT cable is initially connected to the device at such a socket... the outlet cannot negotiate.
That would affect the entire pair, so with 8 also 7.
Conductor pair 7/8 (brown/white + brown) are the same in TIA 568 A and TIA 568 B, so it doesn't matter here. My tip: borrow a Fluke DSP or similar, measure and locate the fault to within a few centimeters. I could help if you are nearby? (NRW / Münsterland)