Does Apple check for copyright infringement in the app store?

Apple does not actively check for copyright infringement afaik.

We're not even talking copyrights... you could take someone's app, copy it verbatim (reverse engineer and reuse assets event) and re-release it with the same name and Apple would accept it.

Apple will only step in if a copyright holder lodges a compliant.

http://www.tuaw.com/2011/02/03/counterfeit-version-of-open-s...

This case was a tiny bit grey, in terms of the licensing agreement. WolfFire lodged a complaint and the App was taken down eventually.

tl;dr; Apple does not check apps for copyright infringement on the App being submitted for review; it only reviews the app upon a copyright holder's request. So don't depend on Apple to protect you from any 3rd party copyright holders.

PS: this obviously does not extend to Apple copyright...

Yes, they most certainly do during review, at least for recognizable branding, etc. The only app I've had rejected was rejected for potential copyright issues.

The rights to the content in the app belonged to the organization I worked for, and the app was being submitted under that organization's iOS Developer account, but the organization still had to prove it had the right to use its own content with a letter from its licensing people to Apple.