Is the re-entry permit desk always open at Suvarnabhumi airport and what do I need to bring?

Re-entry permit counter is at the International Gate 2, before you go through passport control. That gate is the easiest to get a re-entry permit. If you go through other international gate, you have to go all the way to the re-entry counter at Gate2 from air side after you get through passport control. Others details are as mentioned already.

Photo is a sample of the sign at Gate 3, after passport control.

Tell the officer when you go to stamp out in person or BEFORE you go thru the eDeparture gates) you need a re-entry permit

You can only get to the re-entry permit desk if you're flying out internationally, it is open although there might not be someone at the desk and you need to ask an officer to get someone for you

If you have the form, copies, photo you can pay "regular price" 1000baht single or 3800baht multiple

BUT

You can just pay 200baht more (1200baht single/4000baht multiple) and not need anything but your passport 😛