Agreed. On your specific question, the behavior that I was thinking of was like your second option:
In the “Pin…” UI, there is a “Pin to tags” input, similar to the normal tagging field, with the tags currently on the post pre-selected.[1] The limit would be the sitewide-limit for tags-per-post, though.
But I would also be fine with the third (“pick one!”). Worst case, if there’s a need to pin to multiple tags, site admins could create multiple topics (which could even just link to the “main” one).
I think pinned-to-all is the worst case, because it’s likely not right for all tags (depending on how a site uses tags).
There’s an awkward fourth possibility which might be easier to implement: pin to “all tags”, except only allow it when there is only one tag. I say “awkward” because it’s kind of hard to explain and present as a limitation — but practically speaking it would fit my use case.
On your other questions: I’d say: 1. “yes” for tag intersections, and 2. wait there’s a way to show posts by tag group? How? 
Raises question: could you add new tags to the post from that UI? ↩︎