OK, it's too late now, but if you were contemplating going in May, there was no need to send your passport in for renewal in February, if it was still good until August. US Passports do NOT need to be valid for 6 months past your date of travel to Sint Maarten. They just need to be valid through your dates of travel. I don't have any personal experience with getting a passport renewed right now, but I might give it another week or two and then pay to expedite it.

With regard to testing in the US, to go TO SXM, please make sure you read the Sint Maarten travel requirements, which are linked in this post here on TTOL. Link
With regard to which test is preferable, my understanding is that if you have HAD Covid and recovered, a PCR test could come back positive for quite a length of time after, so if you have had Covid, you probably should opt for the antigen test. Otherwise, again, look at the Sint Maarten travel requirements--a PCR test (for US and Canadian citizens) must be taken no more than 120 hours before the time when your last leg of your trip leaves to go to SXM. An antigen test must be taken no more than 48 hours before your last flight. In many places, the results from an antigen test are available in as little as 15 minutes. But PCR tests are considered more reliable. You cannot get your clearance to fly to SXM until you have your negative test results in hand. What I have decided to do, and I think a lot of other people have done the same, is to schedule a PCR test, around that 120 hour timeframe. Since a lot of people have had problems getting accepted, even when their results were negative, this gives more time to get the approval. I will have a backup antigen test planned, in case the PCR test doesn't come back in time. One place I was looking at recently said their PCR tests were averaging between 24 and 36 hours turnaround time, so that is probably where we are going to go.

The temperature checks are to an email address and it doesn't matter where you stay, you are supposed to reply to them, but it appears that this is not very strictly enforced at all, as some people never get the emails and some have not responded to them.

As far as additional French side requirements, that I do not know.