1] Whether or not напиши or напишите is used depends on the situation I suppose. I believe Russian does fall into the category of languages that uses the informal (ты) when an adult speaks to a child, regardless of relation.
предложе́ние literally means pre-pose-al (pose~*ложить), but does mean sentence as well; all good there with cases and whatnot.
I......... would think you would say "write onto something" with на + acc, otherwise you're standing on top of the board while you write. Could be wrong.
классая доска is blackboard for classes-- that's correct. I'm fairly sure I've seen cases where классная is left out due to sufficient context.

2] Я больше не сделаю ничего плохого.
Idunno... Nothing in Russian will line up word for word because of double negatives and the way Russian expresses "again". I can imagine the argument for Imperfective being used here, and it sure seems convincing enough to me, but it may very well be downright wrong.