Just a shot-in-the-dark...
Can all of you install apps successfully, or do they get stuck after the download, but prior to actual install?
Badges on icons that don't increment, or reset, is a symptom of a specific issue tied to a number of other bad behaviors, one of which is the install problem.
Check when you get a moment. If you can download/install, then you are lucky. If not, you may have the same issue that lots of Android users are observing.