I didn't read all of this to see if anyone clarified but it seems a large portion of posters haven't been playing since Heavensward.

*Spoilers for cutscene skippers or people still in ARR.

Hilda is the most famous (or most story visible anyway) half-breed. She is half hyur and half-elezen, which is why the Gentry have given her the title of 'Mongrel'.

Minfilia's 'mother', F'lhaminn, is actually her adopted mother. Their backstories are heavily connected to the 1.0 storyline. She, herself is considered half Seeker/Keeper due to the fact that she has characteristics of both. Also I remember reading that the original Y'da was half Garlean (which is why she wears the headdress, to hide her node) but that could just be fannon or speculation.

The Aesthetician is fully elezen. A bit of his backstory can be seen by doing the Moogle Mail quests. This quest line has some nice race/origin lore bits and its rather easy. I highly recommend it.

Otherwise there are plenty of references to the races being romantically involved, from the Hyur who claims to be in love with Momodi (lalafel) or the weirdest, a Roe man confessing his secret love of Brayflox (the goblin). I can imagine only the ones who are the most physically similar can crossbreed, but the game may prove us wrong.

Also: my theory is that the Twins are also half hyur. Their features are too soft and hilda-like to be fully elezen IMO.