It's true that either a vaccine or test will be and is already needed for many things. It's no big deal, if you choose to be unvaxinated a test will cost you a couple of hundred baht
incorrect, certainly in the case of the father. A birth cert with the father's name on does not confer any parental rights to the father. The OP is wondering if this is the case also for the mother.
I can help with this as I have the same visa extension issued from Phuket immigration for the past 4 years. My understanding is that there is no seasoning required of the 400k and it only has to be in the bank on the day of application however they do ask for it to be in there for a week in Phuket. I have had four 12 month's extensions from Phuket and have had the money in the bank for differing amounts of time for each extension ranging from 2 weeks upto 2 months. The length of time that the 400k has been in there has never been questioned by the immigration officer who incidentally has been the same guy for family visa extensions in Phuket for the past 4 years
Rules have changed now so that unless you are a Phuket resident you can't come unless you are vaccinated and vaccinated people don't need to quarantine so therefore there is no ASQ in Phuket anymore as far as I understand