A 10 foot running start is described in the context of that sentence as moving 10 feet. It doesn't mean you take the run action just because the word run is there.
Is that actually a thing? damage occurring at the end of the round?
If I recall correctly, most such effects (particularly damaging ones) happen at the start of the round, not the end.
Well, the hypothetical character would still be in the air at the start of the round.