Hussar
Legend
Think of them not always as "Half-" races but as "Part-" races and you'll be closer to it; as it only makes sense that when Half- races start breeding with other full or half races quarter-races will result, leading later to eighth-races and so forth. In the game I don't bother breaking it down any further than eighths, but even doing that is way more realistic within the fantasy than just all-half-none. (and yes, we have stat adjusts for each 1/8 level of both Elf and Orc when crossed with Human)
As for their acceptance within the game world, that varies by region and culture - kinda like the real world. That said, the game world tends to be a bit more cosmopolitan (particularly in larger cities or major trade centres) than does ours, mostly because there's simply more races to potentially interact with.
Lanefan
Doesn't D&D, at least later editions, avoid this by having half-races breed true? So a half elf and a human have a baby, it's still a half-elf? I think they nipped that in the bud in the text of the races.
*goes and checks*
I thought I read that somewhere, but, I cannot find it now, so, maybe I'm imagining it.