You can enter Thailand over a land border from Laos if you have a COE, confirmed ASQ and approved transport to the ASQ, plus the Covid test results, insurance and fit to travel document. My Lao wife a British son returned from Laos to Thailand in August.
My 5-year-old has always had an extension as my dependent, even though he goes to an international school. Much easier to keep the school out of it, if you can!
Get the kids non-immigrant O or Ed visas, either based on their school (Ed) or as your dependents (O). Easier to get them as dependents unless you have all the school paperwork to hand.