Is my outside faucet install as bad as it looks?

For what it's worth, I had a GC tell me once that an outside sillcock was 'close enough' and I made the same call you're making now. Made him redo it and it took the plumber maybe an hour. It's way easier to hold the line now than after they've moved to the next phase and everyone's memory of the punch list gets fuzzy. Take photos of the current setup first, then ask for it remounted with proper backing, sealed penetrations, and handles you can acutally turn. If they push back, the photos do the arguing for you.

Agree with the reply about getting it fixed now, and the unsealed holes are the part I'd push hardest on. We had a similar gap around a hose bib at our old place and paper wasps moved into the stucco within one summer, then water started staining the drywall inside during the first hard freeze. Sealing it later meant cutting out damp insulation, way worse than asking for a bead of caulk and proper escutcheons up front. Also on the hot and cold sides, standard practice is hot on the left, so that's not you being fussy either.

On the hot-on-the-left thing, that convention really only applies to fixtures with a single mixed outlet, so a seperate hot and cold spigot can technically land either way, but for the life of me I've never seen a pair mounted hot on the right, and if your wife can't turn the handle at all that's just bad workmanship, full stop. One thing I'd check before they fix the mounting: confirm wheather those are freezeless sillcocks, because if they are, the pipes need to slope slightly down toward the outside so the water drains out of the body when the hose is off, otherwise the whole reason you paid for the frost-free version is wasted.

I'd push back a little on the 'maybe an hour' part of the first reply. If the handles are that tight to the wall, fixing it might not just be remounting stuff. Ours sat flush like that because the stub-outs were too short, and the actual fix was the plumber cutting new copper and re-sweating the connections, plus opening up a bit of drywall inside to get at it. Took two visits. Still worth making them do it, but go in expecting it might be a bigger ask than a quick punch-list item, and get the timeline in writing before the next phase starts.