Certainly for Cambodia they get stamped in and out (Phnom Penh airport was arrival and departure point) but no full page sticker. Can't speak for other countries or land borders.
they can attend school as a dependent too but if they have their own non-ED visa and extension of stay then a parent can get a Guardian visa. One parent per child.
Sorry I'm not American but seeing as I'm also as special as Americans when it comes to visa exempt entry into Thailand, I can confirm you buy ticket, get on plane, go to immigration and get stamped into the country.
Luggage - if your flights are all on the same ticket then yes. If you have two separate tickets then no.
Immigration part - if separate tickets then you need to go through immigration in Bangkok. If on one ticket and you're in transit in Bangkok then I don't know for sure.
no you don't have to, nor should you, spend 3 days anywhere if you cross a land border. Plenty of people cross the border and walk straight back again every single day. The Poipet border is the border where you're most likely to have issues doing that but even then you don't need to, nor should you, spend 3 day there before returning to Thailand.