Nitpicking over whether or not someone's a superhero is silly. Either way, he runs around in his underwear fighting supervillains. And "regular" (I guess) villains such as Anarky (age 12).

I wouldn't call Batman an anti-hero.

All-Star Batman pours bleach over a group of thugs and lights them on fire. I... I feel that's a pretty anti-heroic quality.

Yeah, Batman's no-killing thing is the whole crux of the Joker dilemma, with people claiming that given how many people the Joker has killed and that no prison (or asylum, least of all Arkham) seems capable of holding him, Batman is indirectly responsible for hundreds, thousands of deaths because he won't just end it. And Kaworu, didn't the Crisis on Infinite Earths retcon those early deaths at his hands? It retconned just about a lot of other unwanted things....like Superman's more insane powers.

But then Batman's already out of his head. If he let himself start killing baddies, there's like zero doubt he'd turn into a huge, dark antihero in no time. He'd make Rorschach look stable! That's also the reason that he keeps a Robin around--he's got to have some sort of attachment to the realms of sanity and normalcy, and Robin's it. It's also why fans still tolerate a teen sidekick in Batman when every other hero has lost theirs to age or death or phlebotinum or whatever.

If anything, the most interesting thing about Batman is how hard he fights to not be an antihero.