Rigidity (and height) of drywall around outlet hole
Good to be proactive and catch this before the tile goes up.  Obvious problems I see are that the drywall is poorly cut (can take it out and trace it to re-cut a matching piece without gaps or overcut at almost no cost so easily fixed).  Any of a drywall cutout tool, a Dremel, a keyhole saw, a hacksaw blade or a drill with a side cut bit can cut the piece without so many obvious flaws.  Sloppy cuts like that are no issue on properly backed board, but a repair justifies greater diligence.  If he's being dodgy about fixing such an obvious problem, he may just need an idea of how to do the fix at a reasonable cost.
The rigidity problem won't be fully solved by the above, but it is still very easy to solve.  You have a very significant size workable hole there, so it shouldn't be a problem  to add backing wood verticals to support the drywall.  You probably don't want to remove more tile and trying to screw to the studs inside the wall would be exceeingly awkward, but you can get some 1&quot;x1&quot; pieces of wood, figure out what the longest one you can get through the hole is, cut it and hold it up inside the wall to see if it's long enough to bridge the taller gap with enough overlap to glue to the inside of the drywall above and below.  If it's not long enough, you can put two pieces in the wall and then screw them together with a 4&quot; overlap to make one long piece.  Make sure you drop all the pieces you need in the wall before you start screwing together or gluing in pieces that might block the opening.  Finally, you can cut blocks to fill the space between the backing boards and the back wall and glue, screw and/or wedge those in place.  Consider overkill using heavy duty adhesive like PL400.  By the time you're done the result should be as permanent and rigid as the rest of the wall.  Cost will be quite low compared to cost of re-doing tile an infinite number of times.
Especially if you use high strength adhesive like PL products, prep everything first, clean the inside of the drywall of dust and read the instructions, as some of these products set quite fast.
As far as the depth problem, that should not be a significant issue for an experienced tilesetter or mason unless it is quite significantly proud, but to me it looks inset and since the drywall needs to be re-set anyway it would be easy to shim it flush.
Bashing out more tile, even just enough to install a horizontal backing board would also work, and the 16&quot; tall piece of drywall you're asking about would function too, but it will still have some flex unless you add backing.  You would have to take out 9 tiles including a special cut and really all you want is a normal strength wall and it probably won't break your heart if it has a few extra boards hidden in it.