Banks can do whatever they want. They're a business. You have the choice, keep the money in the bank for 4 months before you need the letter, then you don't have to freeze anything. Or freeze the money for 4 months if not. Or get a new bank. Immigration is irrelevant.
Only the crappy schools that want to dangle the legal status of the child as extortion for the parents to pay tuition. Same as the jobs that won't hire someone unless they switch from marriage to work.
Yes for every immigration. The ONLY things accepted for the first annual extension is 800,000 baht in a Thai bank account or embassy certified income. The US embassy does not do this for many years, therefore the ONLY option is 800,000 baht.
Most doctors will just look at it, laugh and sign it. You might be able to find a doctor that will let you do a video call and sign it for you. There is one in Thailand that will do it that way, but some embassies will not accept it from a doctor in Thailand.
You don't need any visa to study. If you're asking if your child gets permission to stay in Thailand just because you have a visa, then no. You would have to apply for a dependent DTV for them after yours is approved.
Assume you're asking about getting the bank letter from Bangkok Bank. If you don't have the required funds in your account for 4 months already at the time you request the letter, they will freeze that amount in your account for 4 months. You can still use your account as usual, you just cannot access the frozen amount.