Why would the plane keep changing? If there are only these three objects, won't the vectors of their gravitational pull to each other all be on this plane too?
Each orbit is a spinning top. You pull on a top from the side, it's spin axis precesses.
It’s an arbitrary plane, chosen at each moment just so you can flatten it
Not from an external point of view, as you might have a momentum component perpendicular to that plane
(but yes I think you might be right if we're centered on the CG)
If you define the initial conditions such that their relative velocity is zero or parallel to the plane, yes. But that's not the case in general.