Last year I found myself in the vaguely surreal (not really but looking back, a little) situation of sitting on a bus with a group from the Geography Alliance of Iowa in Fairbanks Alaska. A young man from Kansas was explaining to me in great detail, the difference between a Reindeer and a Caribou.
I admit, I wasn't taking it in, I was thinking, 'I didn't know people from the Mid-West could sound as though they come from the south.'
As a general rule, we tend to think that in Canada we have deer - red deer, black-tailed deer (picture)etc. and we have Caribou, and the Caribou are what are called Reindeer in Northern Europe.
Fortunately, the Arctic Wildlife website deals with this, although their opinion seems to be that the two are the same species, just with slight differences in appearance.
Another way of describing the difference is that Reindeer are domesticated Caribou, so you would expect the ones that pull Santa's sleigh to be Reindeer, since they have names, so he must be using the same ones each year rather than just rounding some up from a wild herd.
How about the antler situation? Well, both female and male Reindeer grow antlers, typically the males lose theirs in the late Fall, whereas females lose theirs when they give birth in the spring, but young males often keep them into the spring. Since, from a study of antlers on people's cars, I have noted that Santa's Reindeer clearly have velvet on their antlers, and the velvet disappears when antlers are fully grown, they must be young 'uns, so Santa could be pulled by a gender-balanced team.
And the flying? Well, I go for the generally accepted view. The original St. Nick was from the age that we normally refer to as the Dark Ages, and Shamanism was still widely practised in Europe. The red and white spotted Amanita Muscaria, Fly Agaric, whilst pretty poisonous, was used as an hallucinogen. European Muscaria are also said to be more so than North American ones, why this is I don't know. One of the toxic effects is supposed to be the illusion of flying.
So given the mix of Reindeer, Fly Agaric - nicely dried out from the Fall crop, flying and winter, I think we can join the dots on that one.