Yes it’s possible, though getting a Tourist Visa before travelling will give you more time. You’ll need the money in the account asap. You’ll get an initial 90 days visa then do annual Extensions of Stay.
If you plan to use an agent, then it doesn’t really matter - though they’d prefer the additional time as well. They’ll do the conversion to the 90 day Non O and the first Extension at the same time to give you 15 months.
It’s fairly straightforward to get a Non O from RTE London so that’s another option.
RTE London are usually pretty good about this stuff and know some things aren’t really relevant for a TR.
Passport details page is fine for residency and you can upload the same thing in more than one place. I hadn’t travelled either so just wrote None on a bit of paper and photographed it.
I used a bank statement for my current residency, but anything official with address will be fine. They’ll recognise the children will be with you so use the same for them.
I sent a small packet to Australia and it took 13 days to get from Swansea to Heathrow, sat in Heathrow for 9 days before leaving the country, then sat at Brisbane airport for 7 days before being processed on Wednesday. Waiting for it to be actually delivered now!
If you are planning to live full time in Thailand there are no real advantages to the Non O-A but a huge disadvantage in that it requires specific Health Insurance for each Extension of Stay.
If you are planning to travel between your home country and Thailand on a regular basis then the Non O-A might have advantages in certain situations.