Generally only shifts from adjacent carbons occur (1,2-shifts). Rearrangement from a carbocation on the 5th carbon of 2,2-dimethylhexane would still only result in a secondary carbocation, so there would not be a significant driving force.

Acidity. HBr is extremely acidic (pKa about -8) whereas water and alcohols are about 22 orders of magnitude less acidic (pKa 14-16).

More likely that the shift accompanies loss of the leaving group. You can think of the migrating group as being a bit like a nucleophile that moves over to the adjacent carbon, displacing the leaving group. This results in a new carbocation which can then undergo elimination.

That’s one pathway

Good question. They are known, but only under very exotic conditions. See for example https://pubs.acs.org/doi/pdf/10.1021/ja00326a006

It is much more likely (as you imply) that rearrangement is accompanied by loss of the leaving group in more of a concerted process.

Like many rearrangement questions, these are completely made up.

Honestly I wish I could find more literature examples that correspond to actual exam questions. Then I would show them. They are hard to find.

Rearrangements are much more likely to happen in undergraduate midterm problems than they are in any reaction vessel.

Look for a secondary (or possibly primary) carbocation adjacent to a tertiary or quaternary carbon.

Some of the classic experiments on these shifts were done with neopentyl alcohol and neopentyl halides by Whitmore. https://pubs.acs.org/doi/abs/10.1021/ja01347a037