Yeah well read this : https://fanlore.org/wiki/Alpha/Beta/Omega
But about the first fic... I read before (like two years ago and I can't find it now grrrr) That you can find hints of omegaverse (execpt not called omegaverse) in some Startrek fanfic from back when fanfic was still published in fanzine, so even if its popularity is relatively recent, it has been around for some time before that.
Funnily enough, now the fandom with the most omegaverse is -by far- the Sherlock fandom (like seriously go watch on AO3 : https://archiveofourown.org/tags/Omega%20Verse/works )
Gay, definitely. Witch is technically strait in omegaverse ╮( ̄▽ ̄)╭.
But I've read some gay omegaverse actually, it was rather "nice" ლ(´ڡ`ლ).
I don't like to refer to it as yaoi though. Yes, I realize it's just a term people use to say malexmale but I don't like it when it's used to refer to anything other than... Well, yaoi.
True, in some omegaverse there is nothing like gay or straight. The only stereotyping depended on your presentation. In others, there is some prejudice against being gay or being a male omega, because you would take it up the ass.
As for what you would call it, you could refer to it as slash, m/m, gay, boyxboy, etc... it is really up to what you like using.
I mostly use yaoi or gay as the key search words. If I am desperate enough and there are no results coming up, I actually put in the word 'prostate' along with the other search words. Haha, I apologize. (I don't know why I am apologizing but it seems like I said something weird/wrong, so I will apologize.) ╥﹏╥
Recently, I've noticed a few yaois with "alphas" and "omegas". Is this a new trend or something? If so, when, where, and how did it come into being?