The easiest way imo is to apply for a 90 days Non-O based on marriage in Thailand,then apply for a 1 year extension. But you need 400k baht in a Thai bank account or 40k/month in monthly deposits. You can then work,you just need a work permit.But, you as a foreigner is not officially allowed to work in a Thai restaurant, if that's what you're going for. As a manager/owner maybe,but you can still not do anything in the restaurant, just basically give orders.
Those kids have to be registered in Thailand.It's almost the same routine as being born in Thailand except for the need to translate birth certificates and maybe some other document + I guess legalizations. You normally visit the district office in the province where the mother or in this case the father is registered.
She asked about a service in Pattaya/Sattahip and the office I suggested in Pattaya works fine as far as I know. It's the MFA Consular Services. See link above.
Bank to bank swift transfers are possible via for example Bangkok Banks mobile banking. But I guess there's a transfer limit and the bank wants you to go the branch.
Why doing it so complicated? If you really need to repatriate for example 800k baht from Bangkok Bank, you can simply go to your branch and get it done.