Why Does Peter Rub His Chest in Spider Man 3?

This is to show Peter briefly feeling remorse, before the Venom suit causes him to do something irrational and horrible to someone.

Why does Venom wear a Spider on his chest?

Venom's chest symbol was first seen in the 1980s Secret Wars series, the new Spider-Woman's chest symbol which was then picked up by Spider-Man when he encountered the symbiote that would be Venom. And gave him a brand-new costume.

Where did the black stuff come from in Spider-Man 3?

Having damaged his original costume in battle, Spider-Man discovers a machine thought to be a fabric replicator. The machine produces a black sphere, which then engulfs Peter in black goo, eventually forming his suit while augmenting his powers and abilities. The suit would later be revealed to be a symbiotic lifeform.