Rudolph is a Mutant.He has the X-Factor in his genes, hence being able to emit a glow that can pierce through blizzards yet not blind people.
Hermey the Misfit Elf is actually a human being (albeit a short and/or young one).Hermey differs substantially from the other elves; he has larger eyes, more hair, and, most notably, rounded ears like those of a human. Normal elves may have an inborn desire to make toys (it's a desire that can be either genetic or passed on by magic), but Hermey has no such inborn desire, choosing instead an alternative occupation like dentistry. (He also seems to have no interest in other elf activities like wiggling his ears, singing in a choir, etc.) This could be due primarily to his human heritage. How did Hermey come to live among the elves? Perhaps he was an orphan who was accidentally taken by Santa (like in the movie Elf). Or maybe he was left on the Claus' doorstep. The special doesn't seem to have much interest in his origin story, but a human origin would explain a lot of Hermey's differences.
The doll in the island of misfit toys is there because she is anatomically correct...Unfortunately for her sake, she is an anatomically correct male. This happened as a result of bored assembly line workers deciding to make a doll like this as a joke and see if it got past the product inspector at the toy factory. It did, and as a result ended up a Christmas present for a little girl named Sue that upon receiving it asked "Mommy, why does my dolly have three legs?" After checking under the skirt and seeing that the doll was packing, it was taken away from the girl by mom and dad. After dealing with customer service of the toy company and getting a more conventional doll, the doll with a little extra was thrown away, which lead to the Ruler of the Island of Misfit Toys to pick it up and bring it to the island.
This takes place in the same universe as Toy Story.