, nationality doesn't matter in this case. The embassy letter is not applied for this when the applicant uses the monthly income method for proof of funds while working legally in Thailand with WP.
There is only one re-entry counter at Suvarnabhumi airport and it's located at the International Departure Gate 2, after security screening on upper floor, on the same floor of passport control. The easiest is going through the Gate 2.
If you through other departure gate, you have to tell to the officer at passport control that you need re-entry permit, then you have to go to all the way to Gate 2 from air side to get re-entry permit after you are stamped out.
You can, as long as you are leaving Phuket by international flight(s). The re-entry counter is after security screening. Ask the officer around if unmanned.