If they come in visa exempt they can stay until they are 14 with no repercussions. They can’t be fined for overstay. Until 18 they can’t be banned from Thailand.
When they go to school the school will probably ask for them to get an ed visa but they can actually study without one.
Depends on age of child. Up until 20 it’s possible. Not many other options but look in to Elite if money isn’t an object or the soft power options of DTV if they can show 500k in funding.
There isn’t a limit as such on visa exempt entries. That said you won’t be able to get unlimited entries. At some point the immigration officer will either warn you to get a correct visa or flatly deny you entry.
No one can say after how many entries that could be as it’s up to the individual officer you give your passport to.
If you’ve maxed out two entries and extensions then there is a fair chance you could have an issue.
You’ve posted in the group about the DTV Visa so look and see if you qualify for that. From many reports it’s a relatively easy option and would give you 5 years of entry, albeit limited to 180 days at a time.