I had reserved tickets but it didn't make any difference which entrance it was, the queuing was ridiculous. Basically, they need more staff and also ensure that the electronic self service machines work. There were four of them but none of them worked. We queued for an hour despite booking a slot beforehand. My advice is get there REALLY early!

It is not free any days, ticket is $28 for adults and 16 for kids age 3-12. Only free possibility if you are living in New York, Connecticut or New Jersey. If not, you pay for your ticket. Book online in advance, you need to book a time slot that suits you, they still have timed entry tickets.

I found it comfortable and it was a warm summer day..

Yes

All valid current student ID card gives you the student ticket price. I see no reason why that should not be the case in this situation too.

I had a timed reservation with a ticket and it was just scanned when I entered. This was in Oct 2021 and it wasn't that busy then.

It seems that vax cards will be required. Things are changing all the time so it's best to look at the website and be up to date.

I would ask when you get to the museum. As long as you have a picture ID you should be fine but I would talk with Admissions when you get there.

Usually, I get a timeslot when I go to the Museum directly and see what is open at the time. You can ask the Museum when you arrive to see what is open that day.