A full list of badges with clarifications can be found here: What are the badges I can earn on each site, and what are the exact criteria for earning each badge?, but let me answer your specific questions:
- Shouldn't this description be "First bounty you manually award on another person's question's answer"?
That's Altruist, and the description is indeed rather short; I guess Stack Exchange found "First bounty you manually award (to an answer) on another person's question" too long, since by using the word 'award' it's implicitly implied that you award it to an answer. The "another person's question" part is important to distinguish it from Benefactor.
- What is "First flagged post", flagging others' Q&A or being flagged? Moreover, it seems that being flagged is not a good thing.
That means you earn it when you flag a post, e.g. for being spam, not an answer or very low quality. You can earn it by flagging your own post for moderator attention, but that's rare.
- What does "First rollback" refer to?
See What is a 'rollback'?; it's basically reverting a bad edit to a post. There's a shortcut for this in the revision history of a post (but it's better the way it is now).
Aside, it's good to keep in mind what bronze badges are for:
Bronze badges are awarded for basic use of the site; they encourage people to use all the typical, routine functions of the site: posting questions, answering questions, voting up or down, tagging posts, editing, filling out your user profile, and so forth. Bronze badges are relatively easy to get.
It's important people learn how the flagging system works (it's one of the cornerstones of moderation here); it's nice that users know how to rollback if necessary (manually reversing a bad edit is much harder). Not everybody is particularly fond of bounties, but they're part of the site too and it's good to know they exist.