Yes you can. Most of the people don't have a license. You'll have to pay a small fine if you ride a bike and probably no fine if you drive a car. You can also get a Thai driving license. It is the easiest license to get in the entire world.
For me, when I needed a stamp the lady said no problem after I fix my tm30. After a few hours, I came and the male officer told me to file the 90 days report at the same imigration. So, it is up to the officer.
I would not get that kind of visa with a Muay Thai gym. You might have to pay the full amount for Muay Thai training for a year which is going to be overpriced since they count two sessions a day. Then, if you pay ahead, I am pretty sure the trainers are not gonna care about you. Plus, nobody know what is going to happen during the year. The gyms can get closed again and nobody will return you the money. It's better to get this kind of visa from some self defense school and chose a gym you like and pay by month if you really care about good Muay Thai.