It's impossible to prevent saving the bricks. Preventing the saving of events/ownership is possible, at least for Badspot. But because of how it works nobody can prevent saving the bricks themselves.
Mandatory brick ownership save when on a server seems like a good compromise.
You can still use the build, just not edit it and others can see the original creator.
Nope.
You can still just edit the save file manually, and surely automated tools to do this would pop up pretty quickly. Also, it's rather impossible to enforce that you save it that way, you can simply make a custom script that dumps the brick data down to a .bls, just like it works right now.
Well, it might be possible to prevent the client from saving it, but not for anyone except for Badspot, who has clearly said that he encourages sharing builds.