Received my non-expedited renewal this morning, 13 weeks to the day. Entered online January 4th, went to "in process" 2 days later. My theory is that they're trying to boost domestic travel.
Yes, according to Bermuda. Royal Caribbean says differently, and they're the ones who will be deciding whether you get to board.
https://www.royalcaribbean.com/faq/questions/will-i-have-to-take-a-test-before-i-cruise#united-states
Sometimes they'll get the tenders going earlier than the first scheduled time, so if you're ready early you can get on one without a ticket when they make the announcement.
Do notice that the kits have temperature ranges that they should be stored in. Mine say not above 85F. I'd suggest having them in carry-on luggage, since you can't control what happens to checked luggage, like being left to sit out in the sun.