Here is my two cents:
Paying on time makes no difference as far as I know (I may be wrong)
Taking a class in driving school gives you 10% discount on your insurance on the car that you drive, not the whole policy, so its good idea to do it even if you don't have a ticket.
Klavdiya is correct, going to court is a right thing to do. They will discount your ticket, reduce type from speeding to something like stop sign. If cop or DA is a no show then ticket might be dismissed. In court, they like to give you a $ discount becuase I believe if you paying by mail then $ goes to state, if you appear in court then $ goes to town that your judge and cop is from.
My experience is from NY state although I am now in CT also.
I am surpised that paying it will result in no points. That's explains why we (in CT) have so many crazy drivers. They never accumulate enough points to loose license