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Inflation is supposed to explain this: It could provide the initial impulse to kickstart the velocities. I think the general idea is that the fact that everything is moving away from each other to begin with is explained by inflation, and the fact that this expansion is accelerating is explained by dark energy. Take all this with a grain of salt, here we approach the limits of my tenuous understanding, but what I do understand is that none of this is experimentally verified: No "inflaton" has been found, or any other mechanism to otherwise explain inflation theory has ever been produced such that we could test it, and no working model of dark energy has ever been produced (to my limited knowledge) that we could test or detect.
Tl;dr: I'm pretty sure it's untestable anyway, we basically will never know during our lifetime short of some breakthrough in physics.
Right, but the velocity component would still be present in some form due to the gravitational attraction between bodies. I don't know how significant this would be compared to the redshift value from the initial kick from inflation, or if it is possible to separate the two components somehow.
Gravitational attraction is not a relevant factor on the largest scales where dark energy takes over. To be more precise, it's possible to measure the effects, and to describe a specific distance limit between two bodies where they can no longer become gravitationally bound and are doomed to eventually expand out of each others' event horizons. That limit is the precise boundary between gravitational dominance and DE dominance.
To be specific, literally everything outside of the Virgo Supercluster (home to Andromeda and Milky Way among others) is outside of this limit, and will eventually become impossible to detect because the light between us and them isn't moving as fast as the rate of expansion between us and them. Everything within the supercluster is gravitationally bound, and will eventually (iirc, grain of salt on this one) form a supergalaxy.
Wow. So it's like that adage. However big you think space is, it's much bigger than that.
It's hard to fathom scales at which being gravitationally bound is insignificant relative to those type of effects.
Yes.