Yes, early on Venom/Eddie Brock did a lot of tormenting, even visiting Aunt May on occasion as Eddie Brock, it took her along time to figure out he wasn't a nice boy
his 1st appearance Peter was still wearing the black costume and Venom showed up at his apartment and scared the living sh*t out of MJ (thinking he was Peter until his mouth opened up with his big toothy grin)
and while he threatened it many times, Venom did not eat anyone's brains until Venom: the Hunger when it was explained that the symbiote feeds off of a chemical in the brain that is also found in chocolate (don't remember the exact chemical, my comics are at home).