I imagine both Noxus and Demacia to be rather conservative about the entire thing, not necessarily hostile about non-standard sexualities, but in both of those city-states I imagine the subject to not be something you discuss openly.
In Demacia because of the classic prudishness and some sort of stupid "Their personal buisness is their thing, and if your personal business is non-standard we don't want to hear about it" dogma, so not actually opposed to it or anything, but not open for discussing the subject of even normal cases of sexuality and related issues.
As for Noxus... Well, everything that's taboo is a weakness for whomever has issues with that taboo. So therefore, more taboos mean more people have a weakness, and the more people who have a weakness you don't the stronger you are in comparison. So basically, anything that could be seen as a weakness is taboo because that way it'll remain a weakness, and the people who don't have any taboo issues, or manage to cover them up well enough, remain on top.
On the other hand, in both countries, if you manage to be open about your whatnots, and have the guts and stubborness to pull it off, I'll imagine you'd get a lot of respect about it and wouldn't face many of the barriers society puts up, simply because the systems don't already deal with them at all (unlike ours which have a history of being actively opposed to stuff outside the norm).
Piltover and Zaun... I dunno, I'm not that informed about Zaun, but it seems like these things would be the sort of eccentric traits that anyone with enough money or power could get away with, while anyone without the means to make people accept them would have a harder time with it. Piltover I imagine being more relaxed about everything, because someone would have written lots of papers on the subject and as such a lot of the public (or at least the upper class who make it their buisness to read all the stuffs that everyone writes) would be well informed, and then they'd start teaching it in school and badically, as the city-state grows more informed and people know more, the more open they'd be to 'anomalies' in this kind of stuff because it's not something weird they don't know anything about, it's just another thing that happens.
Freljord: No idea, I imagine them as not very conservative. But not very open about it either. Maybe not very aware of those kinds of issues at all.
The Rakkor: Not that i know more about them than Leona's League Judgement, but they strike me as being very much 'anything goes', if you've earned the right to live past your coming of age, you've earned the right to live as you are.
Bilgewater... who knows. I'd imagine it varies from tavern to tavern and crew to crew... Or even table to table in the same tavern.
Bilgewater doesn't strike me as a very organized faction.
As for Bandle City. I imagine Yordles making a bigger deal of romantic stuff (and non-romantic love) than sexual stuff. So it'd be more common, and any one person wouldn't necessarily have a ready opinion on the subject because it's not as much on their minds as it is on ours.
For the not-diectly-sec stuff, I'd imagine them to be like "that's weeeeird.... shrug well, whatever"
Side notes:
Runeterra in general seems like a very gender-equal setting. Which I like. Sure there aren't as many female champions as there are male champions. But there's enough that you have all sorts of very important and impressive people who aren't men. There's a real streak of diversity in the League, and it's certainly not hindered by the fact that most of the political leaders in the game are (as far as i'm aware) women.
One Champion concept we're really missing is "The Commoner of Demacia". Which should also totally be a woman because really, most of the Demacian champions seem to come from noble/aristocrat families. I wanna see the lowborn soldier, risen through the ranks and having earned their place through merit.
I know it sounds a bit more like a typical Noxus thing, but that's the point. It's how Noxus in general works, which is why it's a more interesting story for Demacia.