you used to be able to do it on their website, I did it on my laptop. I'm assuming you still can but I haven't done one for nearly a year now so don't know for certain if that option is still available.
No one can give an accurate answer as it very much depends whether the immigration officer upon entry to Thailand decides if you're a genuine tourist or not. Chances are you will probably be fine with your plan.
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.