1) It means a document showing you are in the country you claim to be when filing the application. You must file through the Thai embassy where you physically are. So normally you would give the entry stamp from your passport showing that you entered the country.
2) Are you planning on going back to the UK just to file this application? If yes, then you need something with your address, like you say, bank statement, utility bill, drivers license, etc
3) You can start the application on the e-visa portal, but you cannot submit it until you're in the country you are applying through.
4) You will not be on any visa. You will be on an extension. If you do not purchase a re-entry permit your extension will become invalid when you leave Thailand.
non-O is a 90 day single entry visa. You use it to enter Thailand and it gives you just enough time to open a Thai bank account and prepare yourself to apply for the 1-year extension.
Non-OA is a one year multiple entry visa. You get a 1-year stamp every time you enter Thailand until the visa expires. But you're required to go through a much more thorough visa application process including medical check and criminal history check. You'll also be required to have mandatory health insurance to enter Thailand and for the rest of your life in Thailand if you choose the extend based on this visa.
I think they normally want a "certified" translation. Which means a translation company does the translation and then certifies that it's accurate. And then possibly you need to get something done at the MFA after that, but I'm not sure on that part. You should be involved in the translation process if you want to influence the Thai spelling of your name.
Paying rent or not isn't a factor for TM30. TM30 is required by law if you are staying at a location. The owner must file the TM30 within 24 hours. So your wife would submit a TM30 same as before.