The requirements for a CoR vary by immigration office. I don't know specifically what Jomtien asks for, but normally it's as you listed. TM30, passport, possibly rental agreement though I don't think that's normally required at most offices.
The price depends on the office and the time. For example some offices are free or very minimal charge if you wait for 1-2 weeks. But you can try asking if they can do it faster if you need it and they might offer an "express" service for 300-1000 baht.
Many DLT offices no longer accept yellow book for drivers license. That means you'll need to get a Certificate of Residence from immigration if your DLT has this policy. It will probably match the address of your TM30, which should be where your lease is.
the banks say whatever they want to sound correct. Thais do not admit when they are wrong as it would make them look bad which is a huge negative in their society. You can find multiple reports from people who went to their banks and are told straight up lies.
You must apply through the Thai embassy that has jurisdiction over where you are physically. The e-visa application clearly asks for your current location and that's how it decides which Thai embassy will process your application.
There was a recent post about someone who applied at the London embassy and received notice that they had to report to the embassy in London 2 days later.
School age child and affordable are not compatible in Thailand. Any good quality international school is going to cost hundreds of thousands of baht per year.
they don't just close it. They freeze it and you have to go in and show your visa and identity to unfreeze it. If you no longer qualify then they close it.
your inability to open an account has nothing to do with the Thai government. Those are bank decisions to limit their liability. There's no law or regulation telling the banks they can't open an account for you.