I like most of those ideas.
Problem is, if we're orbiting star A, and star A is orbiting star B, then ... here how about a really bad picture...
Facts we know:
Star B has a larger hill sphere (and a stronger gravitational pull than Star A. If it didn't, Star A would not be pulled into orbit.
At some point in orbit, we would be between Star B and Star A.
In order for Star A to orbit Star B, Star B must have significantly more MASS. It will be bigger, probably much, much hotter.
So, when we reach that side of Star A... closest to star B, we would be scorched... All the heat from star A that normally supports life on our planet PLUS a lot more heat from Star B... probably twice what we get from Star A.
If Star B is not that big, it loses mass, loses it's pull, and Star A doesn't orbit.
If we move far enough away from Star B (meaning Star A has to move farther away too,) then we move out of range of the hill sphere (not for a planet, but for a star (A) of any significant mass) and Star A doesn't orbit.
If we're *too* close, or our star is too small/cool, then we get pulled into orbit around star B, instead...
It CAN be done, and maybe something with that could affect something like what we've discussed with the moons, but it would be VERY complicated. Not saying that's a bad thing, it could be fascinating, but I thought I should at least point it out.