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.