I was thinking if you had been hit while down (an auto crit) and then failed the next death save, the reaction might occur after your next failed save. Then I looked it up and it is either or, your choice. So you are good there.The reason to do it this way is 0hps is a floor. You never go less than 0 so if I go to 0 and then suffer more damage it is a failed death save, as long as I do not fail 3 before the start of my turn I will be healed and then can immediately take actions.
The second part is contingency does not state that the character does not need to have the components of the spell. Meaning, do you just get to ignore the verbal and somatic component of cure wounds because you obviously can't do them while unconscious? I guess if you are casting it while casting contingency, then it wouldn't matter.
As reference to the others, it seems perfectly reasonable to have it cast while being knocked unconscious or during a death save. I don't think that part matters because mechanically it doesn't alter the game one way or another.