Car seat info here. Print it out so you have it as ammunition if necessary:
https://www.emirates.com/english/hel...ren-or-infants

Day time from JFK travelling East means the sun will be to your right, but not sure for how long before it gets dark. There are websites that'll track that for you.

Time is sufficient to transit even with a child slowing you down, but don't hang about. Look for any signs at the foot of the aircraft stairs or end of the jetbridge as Emirates will send someone to expedite transfer if they think it's going to be tight. Security checks are pretty lax - no electronics out of bag or shoes off etc.

Not sure about liquids except to say there are rules that mean you must be able to test it if it's over the "usual" limits, so unsealed bottles would seem to be the way to go. Doctor's letters normally mean nothing as anyone could type up a convincing one on a letterhead downloaded from the Internet. Prescription would be better but it doesn't sound like prescription stuff.