Can an STD in your throat cause one side to be sore, one tonsil mildly swollen, with redness on the crown of your throat, and itchy?

Sore throat. It could be especially if there is alot of pus but it is much more likely to be from another bacteria. If it is on one side it could be an abscess and that needs drainage since antibiotics are less likely to work. All and all a trip to the MD is in order.
STDs. There is only one STD that causes pain in the throat and that would be Herpes. You are not in any way describing an STD by the symptoms you list.