Would a 10 day course of amoxicillin for sinus infection cure syphilis or some other std's too if i had them?

STD: No, antibiotics are specific to the type of bacteria suspected as the infective organism and it certainly would not treat viral infection such as Herpes. You need to see your doctor and not just take some antibiotic left over in your or someone else's medicine cabinet.

Only syphilis: Amoxicillin would would cure syphilis if you had it, but it isn't reliable against any other STD.