NaQuela Marie Visa runs (to get a Non-B) are basically your school shifting their work onto you, in most cases. It's more effort for them if you do it in Thailand, and more effort for you if you leave Thailand to do it.
Leaving the country resets the 90 days. I usually report by mail because online hasn't worked since I changed my passport. It's pretty simple and cheap.
I think that makes working here legally very difficult, if not impossible. You'd probably be stuck doing border runs and working on tourist visas, but I don't claim to know much about that either. :P
That's a different thing. You're talking about a tourist visa, and we're talking about visa exemptions. Personally, I have never had a tourist visa converted, so while it may be possible, I can't confirm. That's why I'm just talking about visa exemption (when you come here without a visa), and depending on nationality.
I am not sure about visa types to working visas, but I do know that you can do visa exempt to Non-B without leaving Thailand. It requires a lot of paperwork from your employer, though.