Depends on your child’s passport. If their nationality requires a visa then they need their own to enter Thailand. It’s irrelevant what visa you may hold.
Phuket requires a new TM30 filed even if you stay one night in a hotel or hospital in country (as well as every time you return to Thailand). They do at least have the drive up window so it doesn’t take too long in most cases.
When you renew your passport you take it and the old one to the immigration office that issued your stamp and ask to transfer it. The process should be free.
If you have kids in an international school then they can get education visas and you can get guardian visas providing you meet the financial requirement of 500k (each) in a Thai bank.
You can piggyback off their visas (1 parent per child at school) until they finish school at 18.