I have no expert opinion but here is my non-expert opinion. Hamilton is absolutely a must-see. I would put it first on my list and get tickets soon. It's already sold out through much of the summer. If you can't get tickets there is the cancellation line or the lottery. (Or another poster informed me that they sometimes release house seats a day or two before which I didn't realize.)
Fun Home should definitely be on the list as well.
Fish in the Dark closes in July and I do not believe they'll extend as there is another show slotted for the Cort. I believe OTTC is closing in July as well as others have said.
Finding Neverland and Something Rotten have been available through TKTS so if you're looking to see a bunch of shows, I'd get tickets to Hamilton and Fun Home and then go to TKTS for the rest. That's what I usually do to avoid dropping a TON of money on multi-show weekends. There are people around there who will always give advice as well.... (some paid shills that are obviously not helpful but they're easy to spot). I've always found the TKTS staff to be quite informative actually. I don't follow their advice 100% but they're good at the "if you liked this, then try this..." Type of thing.
Oh, I didn't realize that so many shows will close.. What about 'shows' like Sleep no more or Queen of the night, would you recommend that over a play/musical?
I don't know much about those. I did have a friend who saw Sleep No More and said it was amazing.
If you're looking for suggestions outside your list, Hand to God was my favorite play I've seen in a long time. It's dark comedy. I'd definitely recommend that. Curious Incident is also good. I didn't enjoy it quite as much as Hand to God but it's a good choice and you'll find plenty of people who liked it more (obviously since it won the Tony).
Taye Diggs will be in Hedwig at the time. I adore Hedwig and I adore Taye Diggs so I'd recommend that but with some bias. :)
You won't hear much about amazing grace yet since it's not yet open. Check back in a couple weeks since that's when previews start.