You shouldn't have an issue getting a 30 day extension on your visa exempt entry stamps
You definitely need TM30 showing you're registered at an address in Bangkok, your passport, a passport sized photo. The forms are at the immigration office that you need to fill out and there are copy/photo shops next to the location
For extensions on 60 day visa exempt entry stamps you DO NOT go to the main immigration office at the government complex on Chaengwattana Road.
You go to the IT Square Laksi Plaza immigration office on the 3rd floor.. You can make an appt to show up and be served at this website
as long as it's valid (which it will be until your current extension expires) you can exit/re-enter the country all you want during that time and you'll get stamped back in for the expiration date you have now on your current extension of stay stamp
No problem at all, buy a re-entry permit (either a single re-entry for 1000baht or a multiple re-entry for 3800 baht) at the immigration office where you got your extension issued OR at the airport before you fly out
you can then exit the country and when you come back you'll get stamped in until your current extension expires.