I always had the impression that the Choice was close to the beginning of a race's lifespan. And we know that Tolkien wrote them in for both Elves and Men. The Eldar faced a Choice of whether to follow Orome or stay in the dark; Men...
Actually, I think I can meet you halfway. I don't think the Eldar had a single, unified Choice - Melkor never tried to tempt them at Cuivienen, and Orome showed up before he could. But we know the Noldor had a Choice - whether to listen to the counsel of the Valar, or to race off after Melkor. And, um... they call it the Fall of the Noldor for a reason. ;) So yes, I think the War of the Jewels was the Choice for the Eldar, spread across both halves of Middle-earth - and I think all the Elven races failed, one way or another.
The Noldor, obviously, went psychotic, with many of them actually dealing death to their own kin; House Feanor would definitely qualify as being under the Lone One's control. The Vanyar failed the other way - they stayed aloof from everything, just like the Valar were. There would never be a Vanyarin wizard, because the idea of interacting that much would be anathema.
The Teleri? They had a hard test, with the Kinslaying, and it's no surprise that they retreated into fear of death. But that was a victory for the Lone Power all the same (if not one for the Morgoth).
And the Sindar? I think the Girdle of Melian is their failure. They walled themselves off from the world out of fear, just like their cousins in Alqualonde.
And all of this happens at the same time - in the very short period between the death of the Trees and the First Battle (which was probably around the same time as the Kinslaying).
But what about Men? They didn't have long to wait. Andreth tells the story of how Men fell: by listening to the voice in the darkness, instead of the voice within. They learnt to fear death, to the point where they rejected the idea that they were born with it in place. In YW terms, what they rejected was the idea of Timeheart - the idea that there is a good reason to die out of the world. And except in a very few cases (early Numenor, and Aragorn), they never really remembered that.
I don't think Numenor or the War of the Ring were Choices, any more than the events of most YW books - dramatic though they are - were (though, come to think of it, Deep, High, and Holiday all are...). They were just the actions of the Lone Power in his majesty.
As to the other races: Hobbits are human, according to Tolkien, though I myself suspect some interbreeding with the Dwarves. And Dwarves... are very like Dairine's folk; they're made from rock, remember? They also have the look of a species which didn't particularly succumb to the Lone Power: they live a decently long time, and have at least rumours of reincarnation. I don't remember any of the dwarves we know particularly fearing death. Obviously they didn't pass with flying colours - that only happened once, we think - but they seem to have done pretty well.
And... oh, all right, one more:
Who can we peg as wizards among the Eldar? Elves who do their best to help everyone around them, to slow down the entropy of the universe, and are willing to give their own lives in order to continue that cause?
Well... I know, I know, it's me saying it... but that sounds a lot like Finrod. Y'know, the guy who befriended every species he possibly could - who was the voice of reason in every confrontation between Elves which he could weigh in on - who expressed a deep interest in the details of the Fall of Man - and who, ultimately, gave up everything he had in order to free a Silmaril from the Lone Power's grasp.
And who died in the dungeons of another incarnation of the Lone One. Oh, I bet Sauron was delighted with that...
(Can you have two incarnations of the same Power in the same place? We know they can get as close as, say, Ireland and America - but can they actually meet? Can one aspect of the Lone Power follow the orders of another? I don't think there's a canon answer to that one...)
hS talks a LOT