I was recently served a summons to court for a speeding ticket that was 51 mph in a 35 mph zone. I have been calling the sherriff's office for 2 weeks prior to receiving my summons and they have not even had my ticket on file or in hard copy. The sherriff himself said he did not understand why I was being summoned and I'm really not sure what to even do about it. Can they issue me a summons without sueing me? If so, why do they need to summon me if I have to show up anyway?

A summons is a order to appear in court, for either civil court or criminal court.
I will assume criminal court. If they use the normal ticket, the traffic ticket itself is a summons, to appear in court. Some areas issue a separate summons as a reminder to appear.

Often the case can take time to be entered into computer system, so if there is paper work to appear you just appear

I would recommend showing up or you will probably be found guilty in absentia.