Have to stay in phuket for 14 days before you travel to other parts of Thailand. Looks like you need to take and pay for 3 covid tests during that time too.
I haven't seen anything requiring you to pay 13 days upfront. If you test positive you go to the hospital. Last I read you can stay at multiple hotels during your 14 days.
No idea, when I fly out of JFK I get my ticket from the desk, go through baggage check, and then to my gate. Never seen an immigration or customs officer on the departure side of the USA.