If you married in the UK, make sure you get your Marriage certificate legalised by the UK Legalisation office and then certified by the Thai Embassy in London, before you leave the UK.
Sounds frustrating and yes, anyone with a poor internet connection is likely to have problems. If you use the terminals at the airport it might be easier in that instance.
Depends where you are. If applying for a visa (rather than an extension) it can be in your home bank. If applying for an extension of stay in Thailand, needs to be a Thai bank.