Don't do your extension (1900B) plus a multi-entry re-entry permit (3800B) tomorrow, just get a multi-entry re-entry permit to put you in exactly the same position for 1900B less and virtually no documentation requirements.
Yes, as I mentioned it can be extended before the initial Non-OA expires or, if on a new 12 month entry stamp, before this 12 months is up. Insurance will always be required as all extensions are based on the requirements for the original Non-OA.
You need a new insurance and you can border bounce (no 800k needed) to get a new 12 month entry stamp or you can get the extension if you meet the requirements 800k etc. Whether you get an extension or border bounce just before the Non-OA expires the new 12 month stamp is not mutli-entry so a re-entry permit is required. If you border bounce then you can extend at the end of the second year.
Your school is your employer and your employer has to provide the documentation required for a Non-B application, they may also require you to get your Non-B from outside Thailand. You ask for a travel agent so are you looking for help with your travel to a neighbouring Embassy for your application?
You enter on a new one year entry stamp (if you have insurance) not an extension. Once the Non-OA has expired you will need a re-entry permit to keep your latest entry stamp live or it cancels. You do not need the extension yet.