Build Management is managing the issue fixture tasks in the builds whereas Release management is managing the functionality to be incorporated in the Release.
Build Verification Test (BVT) is done when the build is first received by the testers/testing team. The basic functionality is checked with valid data or valid inputs. This is done to check whether the build is feasible for testing or not. This is done by testers or test team.
End to End testing is also known as system testing. System testing is done by senior test engineers or Test lead.