Using Stack Overflow as an example:
Are there badges that haven't been awarded yet?
Nope Yup. (Precognitive, Constable, and inevitably a bunch of tag badges not shown there if you want to count those.)
Is there a way to see statistics about a selected badge (like number of users that it has been awarded to)?
How can I sort and group badges according to some characteristics (like number of users who have it)?
I would use SEDE queries for this, you can gather all sorts of useful information there. There are also ways to query all sites at once (thanks rene). Note that this information is not quite up-to-date, it is updated weekly.
I have thought about that but there is a lot of badges per site and i think it is hard to do it for all sites manually.
If you'd like to automate this, you can use the API. The
/badges command contains a count for each badge, delivered in a form that can be used by an automated script to run through a list of sites if you'd like to do a bulk query.
For other sites, the same methods are available.
http://site/help/badges will get you to the badges summary, the site can be selected in SEDE, and the API is available on all sites. Newer sites are more likely to have badges that nobody has earned yet, for example, Health.SE has quite a few (missing user count next badge means none have been awarded).