how exactly did rudolph get born

Considering that the filament gives it away as an incandescent bulb, it must have been a hot delivery.

Easy :D This version of the North Pole is a magical land, and it probably has some unusual effects on the creatures living there. I mean, we have intelligent reindeer that can talk, fly, and are self-aware, as well as birds, squirrels, rabbits, and raccoons that can sing and decorate Christmas trees out in the forest. Who's to say something crazily random like a glowing nose couldn't happen in a world like that? Rudolph happened to be the one lucky enough to have such an odd genetic feature. I'm guessing he had a little more magic in him than other reindeer did in that place, it's just that nobody appreciated it until he was needed years later.

He must've eaten yellow snow next to the nuclear power plant.

Disclaimer: I wholeheartedly support Nuclear Power. KNOW NUKES!

His mother and father had sexual intercourse.

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Evolution

He’s a mutant. Professor Xavier recruited him soon after.