Whether or not there was a hearing, that case was filed in court. And, if you plead guilty and pay the fine by mail, it will be disposed in court.
The law does not refer to the defendant being in court; it refers to the case being in court.
If you want to oppose any part of this ticket, consult your attorney.
First of all you have incredibly cheap speeding tickets.
My only guess is that the $50 rate must be something set by the state government. Because that amount is so high, the county decided to add a rather hefty processing fee so they can get theirs.