See GitHub: Stack Exchange, replace badge icons with duckies... and post bug reports and pull requests there.
Install: Install from GitHub
Made some tweaks as follows:
June 01, 2019:
Turned on styling for network-user pages.
Fixed a race condition / GM_addStyle bug whereby the styles were not sticking for certain page loads.
November 16, 2018:
I don't know for sure what the original problem was, but the most recent issue was a result of implementing a new badge without rebuilding the API. Once the API was pushed to production, the bug was resolved. If it happens again (especially if there's a new badge) please let me (or really any employee) know. We can contact a developer directly.
From a ...
The Informed badge is removed from the "Next badge" tracking. Previously it was listed in the Available badges section. Now it is not available under the Available or Bronze section. So the bug has been fixed recently.
Screenshot for reference:
Copied from the MSE answer
This seems to have been silently/magically fixed in the last 7 months (Date of Nicos Karalis' comment).
Descriptions seem correct now. I had verified that they weren't correct when I upvoted this question on Apr 29 '14.
Side note: It's not documented, but apparently tag-based badges are not listed unless they have been awarded at least once.
Take a look ...
Badges and new privileges are part of the /notifications method, which is a single call for all accounts across the network. The results show whether the notification is unread or not. There's also a separate method just for unread notifications: /notifications/unread.
For reputation changes, all I know are some site-specific methods (/me/reputation, /me/...