It is possible to set a 'file' as an user attribute. However, when you set this attribute as required on registration (for new users), the 'browse' button doesn't work. I've seen other posts with solutions that unfortunately do not work anymore with 5.6+.

And is it also possible to make an avatar required on registration via file upload or via the build in flash app?

!! PS When you make this attribute required, you lock yourself out of your site, because you can't select a file !!