By what metrics? He's invulnerable to ordinary gunfire and explosives, sure, but one only has to examine various confrontations with his rogues gallery to see that Superman is not necessarily impervious or too capable.
As portrayed by many writers, yes. That's one reason a lot of readers pick up Elseworld comics like The Dark Knight Returns or Red Son.
Superman in the DCAU is often flawed. For instance, in one episode, an alternate universe (President) Lex Luthor kills the Flash. The Justice League retaliates, and after a brief conversation between Superman and Luthor, Superman kills him.
All these reminders of Man of Steel's major problems has me cherishing the DC animated content even more. Shame the shows ran for so few seasons.