Half-elves do exist, though they are rare and usually mortal. Urgals are compatible with dwarves and humans. Without magic, Dwarves aren't compatible with humans or elves. Dragons aren't compatible with anyone without using magic.
I asked Christopher Paolini this question during his Reddit AMA.
In the books, Arya mentions humans and elves having a relationship, but it wasn't said if these relationships resulted in offspring. Would the child be immortal, like an elf? How about other relationships among the four major species (human, urgal, elf, dwarf)?
They only rarely resulted in offspring. Said children weren't usually immortal. They lived far longer than normal humans, but in the end, they died. There have been exceptions, but they are exceptions of exceptions.
Humans and dwarves actually have the greatest difficulty conceiving children together. Believe it or not, dwarves and Urgals are more closely related than dwarves and humans (remember, dwarves and Urgals have seven toes on each foot, unlike humans).
He went in further detail about other specific relationships in someone else's question.
Can elves and dwarves have babies? Elves and humans? Dwarves and urgals? Urgals and humans? What about once they become riders?
Elves and humans: yes. Elves and dwarves: never without magical assistance. Dwarves and Urgals: yes (they're actually pretty closely related: both have seven toes on each of their feet). Urgals and humans: yes, but rarely.
Later, via twitter, he said that dwarf/human and dwarf/elf relationships were possible.
Can dwarves have children with humans or elves?
Yes. But you end up with some odd combinations of features (plus quite a few possible autoimmune disorders).
I asked him to clarify, and he specified that he was referring to when magic is involved.
But when I asked you this last year, you told me they couldn't. What changed? https://t.co/sl8QnOXsZ6 https://t.co/Bm3oLpEwTb
As I said, it usually requires some magical assistance for dwarves and elves to have children. But it can happen.
Also on twitter, he said that Dragons aren't compatible with any other species (unless magic is involved).
Could a dragon mate with another species for some kind of Chimera/griffon type creature?
Not without a great deal of magic. (Stop thinking about it. Seriously. Stop it.) :D
He went more in depth about human/dragon pairings in his second reddit AMA.
What if you had a messed up human/dragon pairing that attempted to mate, possibly with the help of magic/body modification? (like how the elves can modify their features) What kind of a creature would come from that? A giant human-ish creature with wings, scales, & fire-breath?
Yeah, humans and dragons can't mate. Not without magical surgery. Stop thinking about that. Seriously. Stop. :D
Assuming the involved human and dragon had their required surgery/magical physical modification/the relevant biological materials were transported via magic (like Saphira's egg), what sort of Cthulhu-esque monster would we get?
Seriously, how would a human/dragon chimera (or something of that type) affect that world, and how many of the already-existing groups (governmental and magical) would want to destroy it VS recruit/train it?
Hmm. Well, even if you could get the relevant biological materials into the right places, you'd still have a chromosomal disparity between human and dragon. If that could be overcome (which is a big, BIG if), you would end up with a being that contained some combination of human and dragon features, many of which would be inimical to life. But if it did happen, and the being could live, I'm sure it would be regarded as the oddity it was. The creature would probably end up living by itself or with the elves, I would imagine.