Well, it turns out that you're in luck! We have such a site where you can do exactly that. It's called the Stack Exchange Data Explorer, and it allows you to run queries on the public data dumps.
Many of the queries that have already been written and shared focus on evaluating your progress towards particular badges. And if you can't find the query you want, you can write your own.
For example, this one shows you the progress you've made towards earning the
As far as integrating these features into the site itself (e.g., seeing the count when you hover over a badge, as you mention), that's unlikely to happen. Jeff has said a couple of times that badges are supposed to incite curiosity and it won't always be immediately obvious how you've earned them.