The only thing the alliance affects that I know of is who you fight for in PvP/Cyro.

During PvE it has zero effect.

As for LGBT guilds I have seen a few in game but I don't remember the names. One was something like "shot in the knee with an arrow"

Edit : AFTER I wrote all of the below, I realise you're on PS4, so that won't help. But for everyone else wanting to know, I'll keep my post.

Chimera Syndicate - An active, and engaged LGBT + family friendly guild with a 24 hour rolling average of 30-40 players. They do trials, overworld content, cyro, and throughout the 2-3 years of me playing on and off, I've never not seen @Gracefully_Lyris and the team not have a weekly organised calendar mailed to members for upcoming events.

I actually ditched a few other guilds I had because they always seem to have a trader out of Senchal for me to sell on, and the guild hall for @Evil_Dave literally has... all stations, and every facility.

Havengarde - Another active LGBT guild run by some very enthusiastic players, and probably a bit more chatty and interconnected, but not Cliquey. They've generally got an average of 30 players at peak, and 10-15 off peak (as I am in Australia, so I generally see the changing of the guard with player hours), and to be honest, I adore this guild.

Doesn't have all the features of Chimera, but being in both communities, which funnily enough doesn't seem to have much cross over in members, has been a nice experience.

I know there are a couple other ones out there, but I don't recall their names.