If they are flying away, something would need to add kinetic energy to the stars over time. Also, galaxies should then start small and disperse as they age.
not exactly. The gravitation gathers the stuff closer, and at the same time the rotational momentum must be preserved. So initially that causes the galaxy matter into elliptical shape and from elliptical galaxy the disk forms and is thinning and spreading further to preserve the total rotational momentum of the galaxy compensating for the loss of the momentum by the matter gathering at the bulge. The radiation pressure seems to be a major component mediating that process (and in particular causes and supports the push away of the stars from the bulge&bar - part of the kinetic energy addition you mentioned).
This is why we don't "observe" that much DM in elliptical galaxies as that momentum spread by means of forming the disk is only starting there.
It's not the rotation that assists, it's the planet's linear -- well, bent, but that doesn't help -- that does it.
There is this notion of "frame dragging" in general relativity, but its effect is practically impossible to measure. They lofted a satellite to measure its effect over several years because it took that long to accumulate enough to measure.