You'd be wrong though. Evangelical Christianity holds sway in the US.
And they make up something like 26% of the population, and as such are mathematically NOT A MAJORITY. You do not have to have a majority to hold sway. Hell, you don't even have to have a lot of followers. Just having money can give you sway.
Daenerys occasionally worshipped the Seven and the Dothraki Horse gods and married one of the latter's most prominent worshippers, explicitly rejected R'hllor as a god, and her chief advisor was the Green Grace, head of yet another religion.
All that made absolutely no difference to Benerro,
Irrelevant. You'll note that none of that involved actually declaring R'hllor or his followers her enemy, or marrying a man who did so and giving him support. The likely best case for Dany in regards to Benerro is that he still things Dany is Azor Ahai and needs to be rescued from our foul clutches.
R'hllor isn't someone whose opinion on the matter we've heard, other than it being obvious that Dany isn't actually Azor Ahai so he might just not give a shit, provided the Melisandre's visions are actually from R'hllor at all.
I don't like Cthulu either. Sue me.
Irrelevant. A priest explicitly sent to Daenerys by Benerro himself holds that this other god he probably knows very little about stated that it was a demon. If you're going to send someone as ambassador to the person you believe is your religion's messiah figure are you going to send someone whose views are radically different than your own or are you going to send someone who is firmly in your ideological camp?
Melisandre is a very unusual Red Priest, given that she's from Asshai By the Shadow and is a Shadowbinder, unlike the priests from Western Essos that had zero problem with coexisting with other religions for over a thousand years without a single hint of any religious conflict. She's also in desperate circumstances, given she's staring the apocalypse in the face. We know that sacrificing the icons of other religions gives your god power. Why do you think she was doing anything different to what we did to Drogo out of what she considered necessary?
Apart from the fact that no Red Priest from the Free Cities ever tried at any point in any circumstance to suppress or initiate conflict with any other Essosi religion, even the smallest and weakest?
What you propose is clearly false.
General freedom of religion in the Free Cities has existed for a long time, and trying to go against that would probably not be taken well. The precedent of religious tolerance was established by the dragonlords, since religious diversity kept their subjects divided. (AWoIaF pg 257) The actual rulers of the Free Cities probably maintain a similar policy for the same reason, and would probably crush any religion that would try something so brazen - if one city fell to that, the others would probably crush the cult in their own borders, if they were able, to prevent the same thing from happening to them. Outside of something rare like two religions having a spat due to an incident, they're expected to just leave one another alone. General inertia probably also factors into it. I wouldn't be surprised if R'hllor's priesthood is willing to take the long view on things, slowly gathering more power and followers for the eventual coming of Azor Ahai, and they may not have been nearly as powerful for most of the time since Valryia's fall, if they were even around for that long. (religions do rise and fall in Essos, so they may not have been around for a millennia)
I wouldn't be surprised if the general worshiper of R'hllor isn't generally informed of the view that other gods are demons, as some cults don't give you the "full story" until you're higher in the ranks and fully invested. Alternatively, it might be a relatively new view on things that is spreading amongst the priesthood as their religion has gained greater power and prominence - religions aren't static, after all, and power does corrupt.