If you are doing online coaching and earning money from outside Thailand rather than working in person and earning baht then I believe that would be OK.
need it for work permits or teaching license. Can't remember exactly which office you need it for because it's all just a bureaucratic hassle that blends into one.
Can't remember exactly the building but if you show up around there and ask you will find it easily. You may have to book an appointment in advance. I had to once but not the other time. This is the website