You might wish to have a look at git-annex: http://git-annex.branchable.com/

The answer in any case is: yes, you can. However, the next question would be: "How?" and "Is it worth it?".

I haven't used git-annex yet, but it seems to me, that might be the solution for you. If I wanted to share a lot of big files, I'd certainly give it a try.

What kind of sharing are you aiming to do? How big is your collection, and how many people are you sharing it with? If you'd like something to synchronize directories, you could use rsync. But you'd have to be more specific.

As for uses other than source code, there are some wikis which use git as their storage back end, for tracking versions.

Technically, I guess you could, but this would be a quite a long way from what Git is good at - text files. Git does a bunch of compression of changes that works well for text, and doesn't work at all for binary files. Your pulls and pushes will be very large.

This is probably possible, but not a very good idea.