They both can be used, but can mean different things. If it is one session of drawing instruction, you can use "class" or "lesson". If you are referring to an ongoing series of sessions, you must use "class" or maybe "lessons."

I would say in school situation " drawing class". If you had a special session outside of that class then I would use "lesson". Example My daughter has an orchestra class in school (all students in orchestra). She has a orchestra (violin specific) lesson later in the day special for more one on one time for help with her playing ability.

Yes.