1. I was wondering what sort of things the standard search bar (see image below) is programmed to look for. Does it look for words, phrases or names in posts to narrow the results?
It's just a fulltext search on a post message. So it's primarily useful for posts where someone actually typed something and didn't just upload a photo with nothing else.
2. For the @mention, I was noticing the results pull up all members that meet the criteria (such as first name), but in the case of a Group Stream, shouldn't it only show results for people in the group?
Mentions is just for mentioning anyone. It does not filter based off the stream it is on so no that's not expected behavior to only be able to mention those in the group. The only control over this is the parameter in CB Activity > Parameters > Streams > Mentions > Who. There you could set it to Connections so people can only mention other users they're connected with.
3. For my main feed, only paid members of a certain user group have access. I would prefer not to have the free members show up in search results like @mention since they don't have access. Is this configurable?
No. Anyone can mention anyone. If you don't want free users to be able to use mentions though that is doable by adjusting the mentions display access in your streams settings under the Display tab. Search and filter on streams can also be access controlled and you should see parameters for that under the Filters tab of your streams.
4. I wonder if AI would be a good tool to inject into the search function, and if that is something being considered in future development for activity feed search?
Not sure how AI would help with searching. With posting though there's possible future plans for AI assistance as well as AI live translations of posts. It's possible to integrate AI now though as CB Activity has a lot of triggers for easily adding new features, but you'd need to be familiar with PHP at the least to get started with that.