In my campaign:
-- Elves & half-elves v. half-orcs & orcs can't interbreed. It's a rule of the universe. (The creations of enemy gods.)
-- Dwarves interbreed with no one. Ditto on Gnomes.
-- Goblinoids can presumably interbreed, but you'd likely just get more of the weaker version, as in goblin + hobgoblin = goblin, goblin + bugbear = goblin. Can goblinoids interbreed with orcs? Probably. Can trolls or ogres interbreed with others? Nope.
-- Halflings are actual the middle ground -- Tallfellows are part elvish, Stouts part Dwarvish, and Hairfoots part gnome. Presumably they can mix with human too, resulting in short humans? Probably can't mix with orcs.
-- Mongrelmen, who can explain their origin, but they do exist. Magical experiments? Magic runoff contaminating the drinking water?
-- There is no such thing as half-dragons. Polymorph doesn't make it possible. It's just another rule of the universe, as in the DM finds the concept annoying.
Halflings: Don't exist. See fey.
Orcs: Don't exist. See goblins.
Gnomes: Don't exist. See fey.
Fecund races
Humans: Can interbreed with fey, goblinkind, and elves.
Goblins: Can interbreed with fey and humans.
Elves: Can interbreed with fey and humans, although for a variety of reasons this actually happens quite rarely. There are no half-elf populations, and at any time there might be fewer than a dozen half-elves in the world.
Fey: Can interbreed with just about anything.
Non-fecund races
Dwarfs: Can interbreed only with fey, although this happens rarely.
Orine: Though they find humans and elves mutually attractive and dalliances are frequent, can interbreed only with fey.
Idreth: Can interbreed only with fey, although this happens very rarely.
Dragons: Like fey, could potentially interbreed with just about anything, though in practice almost never successfully and in general, they don't find anything but another dragon to be attractive.
Genii: Like their close cousins the fey, could potentially interbreed with just about anything, though mere mortals are generally considered by them to be beneath them.
Other: All other humanoid or demi-humanoid races are creations of individual deities, and designed to be servitors and further that deities interests exclusively. For example, Kobolds are not little dragon people, but the rat-kin servitor race of a deity called Sormkortek who they serve and worship more or less exclusively (though kobolds are relatively unusual among servitors for practicing, with permission and encouragement of their patron, ancestor worship as well). In general, they can't breed with anything else that isn't also a servitor of the same deity.
Giants: Offspring of the gods and the genii. Like their ancestors, could conceivably breed with just about anything. Physical limitations aside, generally have no desire to do so, though its believed that the race called Hill Giants are mixed giant/human blood and that Ogres are mixed giant/goblin blood.
Aberrations: Created in experiments by a infamous group called The Art Mages - basically, gated in from outside the universe.
Beasts: Another different set of experiments by the The Art Mages - basically, genetic engineering.
Magical Beasts: Most are akin to the concept of native outsiders. Can't really breed with anything but each other, and in fact many are spontaneously generated at least as often or more often than they are results of breeding.
Undead: Can't breed, though can reproduce by killing things.
The above is complicated by magic, which potentially allows almost anything to happen, and deities and similarly potent outsiders which can breed with just about anything and thereby leave pseudo bloodlines behind. For instance, a deity like Ishi could breed with an evil drawf, resulting in a linage of dwarves with divine blood and goblin blood traits even though no actual goblins were involved and even if Ishi did not take on a goblin form. Thus, the most likely parent of a half-dragon is almost certainly the Mother of Monsters, Tiamat herself, who is not nearly as limited in possibilities or attractions as her offspring. This general approach is in keeping with some of the literary inspirations of my campaign world, such as the Greek myths, which make demigods (in the sense at least of being offspring of the gods, not in the D&D sense of capable of supporting clerics) not all that rare or unusual. At any given time, there are a lot more half-gods roaming around than there are half-elves, and most of these are not Divine Rank 0 beings (much less Divine Rank 1).
For any sort of exotic parentage, generally I'd choose an existing racial type and encourage the player to take one of my existing parentage/adoption background traits on character creation.