you can prepare the application in Thailand but you can't submit it until you are in Laos as you need the entry stamp to show your location, but you always have to be in the location you apply from in any case. Once submitted you go to the consulate and pay
you need the supporting documents from the employer wherever you apply as well as proof of relationship but you don't have long enough on your current stamp to apply inside Thailand so you would have to leave anyway
There's normally a difference between cancelled and rejected. If your application was cancelled I'd say that is due to incorrect primary information. The name you provided not matching the name on your passport or something similar. The passport scan is known to be problematic so unless you triple checked that it would be where to look
I think your a little confused. You can get almost 2 years out of a Non-OA visa by exiting and re-entering Thailand just before it expires as long as you have another years insurance. You can get an extension of stay based on a Non-OA visa indefinitely but you do require insurance every year