Half-human half-elf is a race mentioned in the D&D handbook. Is there also a half-dwarf half-elf combination? And would that be common? If it doesn't exist, how could I create the stats for it myself?
Traditionally,1 humans (and dragons2) in D&D are just... amazingly, maybe magically, compatible with other races. Massive numbers of them. Any time a race is “half” something, the other half is human 90% of the time. And there are numerous races that are human hybrids without “half” in their name.3
For the last several editions at the very least, half-elf has been a core race, and half-orcs were too in 3rd, 3.5, and 5th (and 4th got them not long after, in Player’s Handbook 2). In all cases, the other half was human. In fact, in none of these editions were there any elven hybrids with anything that wasn’t human (as nitsua60 points out in comments, orcs could also hybridize with ogres to create ogrillons, a very rare case of a hybrid with zero human, dragon, or outsider blood).
And during this time, dwarves only ever saw one hybrid: muls, which are specific to the Dark Sun campaign setting, and are again half-human.
So for whatever reason, in default D&D, humans are astonishingly capable of hybridizing with everything else, but none of those they can interbreed with seem able to interbreed with one another. While not... strictly impossible, I suppose,5 biologically this is extremely unlikely. Then again, so are a panoply of independent humanoid races, so meh.
It is up to your DM whether this fact is canonical, or just a quirk of the material they’ve chosen to publish and not representative of the world you’re actually playing in. To my knowledge, there are no explicit statements that elves and dwarves can’t hybridize with things other than humans, they just never published such a thing.6 So your DM could easily homebrew something. For that, nitsua60 has already pointed you to the correct place: Dungeon Master’s Guide page 285.
I am not aware of anything prior to 3rd edition that contradicts anything I am saying, but there is a ton of material, particularly for AD&D, that I have only passing familiarity with. Particularly the Planescape setting is likely to have something. I am researching this possibility, but for now assume all of my statements apply only to D&D editions by Wizards of the Coast.
You can apparently make a half-dragon with anything that’s got a living body. Half-dragon animals are not even all that unusual in the books. Half-dragon oozes are even possible. Many of these are chalked up to magic rather than natural reproduction, and even in the cases of “natural” reproduction they’re facilitated by dragons’ innate polymorphing ability, but still.
For example, the original Eberron Campaign Setting rulebook for D&D 3.5 introduced four new races – and three of them were human hybrids (changelings are half-doppelgangers, shifters are half-lycanthropes, and kalashtar are half-dreams3.1). The Dark Sun campaign setting has half-human, half-dwarf muls. Forgotten Realms has got a bajillion elven races, and many of them detail their own special half-human counterparts.
- Yes, really. They are true-breeding hybrids of humans who voluntarily allowed dream creatures to share their body and soul, and are now born with two souls, one human, the other dream monster.
D&D 3e and 3.5 had a concept called “templates” that could be applied to a wide variety of creatures and races; as hinted above, half-dragon was one of these. So you could have a half-dragon elf or a half-ogre orc or whatever, since half-dragon and half-ogre were implemented as templates. No template for half-elf or half-orc was ever implemented, however, so you couldn’t do that with them.
Please don’t start quoting that thing you think is the definition of “species” at me – there are numerous edge-cases that make “able to produce a viable offspring” fail as a definition of species. In fact, defining the word “species” strictly is extremely difficult and there is no scientific consensus on a precise and strict definition for deciding when two things are in separate species.
KorvinStarmast reports that in 1st edition AD&D DMG and PHB state that the world is human-centric and that half-elves and half-orcs are the only half-breeds. This is obviously not true, since there’s plenty of others (that are also half-human, and a very few that aren’t), but it does give some weight to the idea that this is not just a quirk of what was actually published.
There isn't a canonical Dwelf option. In my experience that would be fairly uncommon, but it's nothing that hasn't been contemplated. Race is a tricky matter, even in D&D.
But you can make one. The Dungeon Master's Guide, starting on page 285, has guidance on factors to consider when creating new player races/subraces along with two worked examples. You could also look at the player's companion from Adventurer's League's Elemental Evil campaign for other examples: Aarakocra (bird-people), Deep Gnomes (Svirfneblin), Genasi (half-elementals), and Goliaths (really big people).