1. As someone already said, Dr. B. supposedly has a chip implanted in
his head so that he can see/her her. But yes, I also think he's simply
going insane, though she may be actually there.
2. The 30 minute clock, as I understand it, was how long it took the
Cylons to track them down. Somehow the Cylons were able to find them
after each jump; thus the days of jumping continuously. By the end of
the episode it's resolved why they were being tracked so easily.
3. Distance is never really explained, but the jump is planned.
Again, we're not told how or where, but I think it's safe to assume
that Adama is either plotting the course to "Earth" or towards the
largest star cluster of unexplored space he thinks they can get to.
4. I think mostly because of two reasons: First, a female Starbuck
provides a much more diverse starring female cast--the first series
didn't focus on women much at all, though there were a few. Second,
it's more interesting with Starbuck as a woman, frankly. You get to
bring her in as a real star character that isn't Apollo's wingman;
she's her own ace, her own pilot. In that regard, they're able to do
more stories centered around her, create more complex relationships,
etc. That comes into play in the later episodes.
5. Shaky cam is a (relatively) new Hollywood trend. You could also
argue that it provides a more tense and chaotic scene.