Is there a "Lightroom for Video" product?
Short answer, there is not really any "Lightroom" for video. But there are non linear editing programs out there that do all the things you want to do with your footage. So, as someone else said here, you want to start using Premiere or some other editing program. 

Learn how to organize your projects professionally. You're never going to make duplicates of files (except in some circumstances), but you are going to create duplicates of edits, string-outs, sequences, clips, etc.

So, here's a typical project organization, inside Premiere in this instance (make a clean version of this project, call it "Default", and simply open it and save it with another name each you work on a new project; that way you don't have to re-create all those folders all the time):



And here's an example of drive organization: