A couple of observations:

(1) JKR has said that Teddy is not a werewolf, even though his father was one. So lycanthropy is apparently a magical curse, not a heritable condition.

(2) We don't really know how vampirism works in Rowling's world. Since she tends to more or less follow traditional legends (usually), we can assume that vampires are, in fact, undead, and thus incapable of having children. But Rowling has never explicitly said that, so there isn't an official canon answer.

(3) As a couple of others have noted, Rowling has said Flitwick has some goblin ancestry, which means goblins can interbreed with humans. Several people have suggested elf/goblin crossbreeds. I would suggest that this is even less likely than goblin/human. You're assuming that because they are similar in size, they might be more similar genetically, but from what we've seen of their traits, goblins would appear to be much closer to humans than either are to elves. Goblins and humans can both do magic only with wands or other tools, whereas elves seem to be innately magical beings. Elves appear to be much more alien, whereas goblins, while different from humans, have lifestyles and physical attributes that are much more recognizably human.

(4) Remember that faeries, according to FB, are non-sentient creatures, little more than humanoid insects, not the ethereal, magical beings of legend. Ditto imps and pixies. So you'll have to make some other critters responsible for changelings if you want to introduce them in the Potterverse.