Definitely looking good. It's evident that you've done a lot of research - you've also received a lot of great input here.
I'd definitely agree with the temp recommendation. We run our tank at 78F-79F in the winter, closer to 80F-82F in the summer. You'll find as you go on, that as much as you can mimic natural conditions - this will be to your benefit.
Phosphate, Calcium, Magnesium, Iodine or Strontium yet
Calcium is probably one of your more important tests (
if you want to keep corals). You'll want a phosphate test if you ever run into nuisance algae issues.
I personally have strontium, magnesium, & iodine tests, and use them maybe once a year - IMO, I would not waste your money on these. I'd suggest these only for tanks with a lot of coral growth, and I would only recommend buying them if you suspect something is wrong, or if you plan on using any additives.
So now what? I know the nitrate should be lower.
I'd start with a water change - around 30%. You probably hit a "mini-cycle" due to die-off from the move. Nothing to be overly concerned about.
I'd say you could go ahead an add a cleaner crew. Research this carefully, you probably don't want a pre-packaged "kit" - it'll likely include things you may not want. You'd be better off to mix and match a few different snails, and a couple other reef-safe critters.
HTH, & good luck!