What if one hosts a code repo in the Dropbox folder and shares it with others who collaborate. What happens if two people push to the Dropbox repo at the same time? Could this cause conflicts that mess up git?
Dropbox is not the right kind of hosting service to share Git repositories. You can get all kinds of trouble in the moment when a conflict inside .git/ occurs (and such conflicts can even occur in a unchanged repo, for example when one runs git gc), since the content of the .git directory is not designed to be easily merged.
You can use https://bitbucket.org as an provider when you don't want public accessible repos.