... there were no female Dwarves?
... there were no male Elves?
I quite like the idea of Elves and Dwarves being the somewhat sexually-dimorphic representatives of a single species (so the offspring of a Dwarf and an Elf would be an Elf or a Dwarf). Of course, assigning the sexes other way round is also possible, but 'The Dwarf-King' and 'The Elf-Queen' are probably more mythologically-grounded in the Western canon than the other way around. That in itself may be reason enough to insist on bearded Dwarf women and willowy Elf men, however.
There is no reason why this can't work with the standard B/X rules. There's nothing in the rules to say that Elves and Dwarves have to have stable nuclear families of their own species. There's nothing really in the fluff that says anything about Elven and Dwarven families either, but the point of the experiment is to remove the fluff anyway and just concentrate on what the rules tell us. The fluff says that Dwarves "often live underground", whereas Elves generally "spend their time feasting and frolicking in wooded glades" (B9). Both races enjoy feasts and exquisite craftsmanship. It's starting to sound to me like they may actually be the same race after all...
... the offspring of a male human and an Elf (always female) was a Halfling (of either sex)?
... the offspring of a female human and a Dwarf (always male) was also a Halfling?
Halflings, being either male or female, and en masse both male and female, work fine as a stable population in their own right. So there's no real reason to alter them, I decided. But, there would probably be Halfling communities near all of the others, and many Halflings living in and around Human, Elf or Dwarf settlements. Maybe fewer around Dwarf settlements, but still there perhaps. They make sense as the somewhat-odd offspring of these 'fair folk' and humans. After all, they look somewhat like Dwarves, and act somewhat like Elves... in fact this whole conception started with a desire to make a B/X campaign where people could play 'Half Elves' and 'Half Orcs'. I decided that if people wanted to play a race that was somewhat arboreally-inclined and good with missile weapons, a bit Ranger-ish, they should play a Halfling, as Halflings are the Rangers of B/X. By the same token, Dwarves are the Barbarians of B/X, and if someone wanted to play a Half-Orc (a race I believe that is often used as a basis for a Barbarian) they should play a Dwarf, ie an angry hitty thing. I came up with a chart that looked something like this:
But, taking this basic idea (that the races are just blendings of other races), I then played around and decided that male Dwarves and female Elves as members of the 'Fair Folk', and Halflings as 'Half-Fey', might actually make more sense.
I do not propose to run this as a campaign. It is more of a thought experiment at the moment, part of my poking into what B/X can actually be made to do without doing any violence to the rules.
An alternative D&D setting
Keep Humans but scrap the other three PC races. Elves, Dwarves and Halflings, as presented in B/X, are pretty much tied to tropes inherited from Lord of the Rings, and beyond that to various bits of European literature like the Morte d'Arthur. Instead, non-Human PCs can be Chandali (tricksy, nimble forest-living folk, a bit like non-shit Ewoks), Voorn (mostly tough angry fighty dudes with some construction-related skills - I imagine them as somewhat rocky and trollish) and Jadarath (tall noble blue-furred tiger-people who combine fighting and magic).
In other words use exactly the same rules, but change some names and descriptions to get a different take on some familiar fantasy tropes. There's nothing in the rules to connect Elves with forests per se, or Dwarves with mining and smithing. So re-skinning them means cutting away the 'cultural baggage' derived from Tolkien and re-imaging what the rules could refer to in a different setting.
In this new setting, it's the Chandali who live in the forests, in Ewok-like villages high in the trees. The Jadarath live in towering cities wreathed in cloud and are something like Furry Jedi seen through the lens of the Mahabharata. The Voorn, I'm not so sure about yet, I need to think about them more, to find ways of not making them 'Dwarvish'. They can still all fit into the Quest for the Relics of McGuffin, which I take as being the default overarching plot in so far as there is one for D&D.
I've written a little before about arranging a journey-based game with a changing party here and I think that the structure holds up. But it does mean that the players have to agree that 'the quest is the thing', as I have discussed recently. That, in turn, comes back to why the PCs are adventuring. But this post is just about different ways of considering 'where' - especially if it doesn't have to be a standard Tolkienesque setting.