Don't know which state you live in, but here in Cali class C is for cars and class M is for motorcycles
What license? A standard drivers license is no good for motorcycles. You need an additional license for them.
There is no distinction between a street-legal dirt bike (dual sport), and a regular bike. Here (BC), you can ride a scooter or a moped on a car licence, but not a motorbike. I'm not sure what the legal definitions are, and it's you that has the problem not me, so look it up for your state.