Well, this is a Discourse forum, so about Trust Levels, this FAQ should answer your questions
https://meta.discourse.org/t/what-do-user-trust-levels-do/4924
As for Badges, (the ones that are not associated with a Trust Level) there are no “extra abilities” that come with getting them. They are more like a recognition of “milestones” that have been reached.
Note that some of them can be used as a “Title” (see your Profile Preference settings.