Question: How does IE register ActiveX controls for use in the browser?

Does it just run regsvr32 for the DLL?

Answer: 
ActiveX components register themselves, triggered by a well known DLL entry point (`DllRegisterServer`).

`regsvr32` is just a wrapper around loading the DLL and calling that entry point. Other tools can do this directly. Installers sometimes just directly update the registry (having recorded the changes to make when building the installer).
It actually doesn't have to do any of these things; the CAB file specifies what it will actually do. It may use DllRegisterServer, and indeed this is the most common thing, but it could also launch an MSI or EXE installer that may register the ActiveX control in another way.