1) You are by no means alone in playing a character of a different gender than your own (it's actually relatively common, even accounting for stereotypes regarding people playing catgirls, etc.), and pretty much no one will actually care unless you make a big deal out of it yourself.

2) Summoner gains access to 'Egi' versions of some of the classic FF summons you fight in the main storyline - Ifrit, Titan, and Garuda. You also can summon a version of Bahamut at level 70 and once Shadowbringers drops, you'll be able to do the same for Phoenix. I don't play with a controller, so I can't speak to the difficulty of control, but especially in casual content you don't need to worry too much about micro-managing your pet, especially as a summoner; setting Ifrit to Sic and letting it do its thing is pretty straightforward.

3) Free Companies - this game's equivalent of guilds - are mostly social communities rather than devoted specifically to running content types, and there are plenty that mostly just act as places to hang out. There's a subreddit (r/FFXIVRECRUITMENT) and subforums here that are good places to look for guilds recruiting new members, and of course you can always end up joining one just through people you meet in game.

Also, don't feel bad about being a casual player - you're actually part of the majority who play the game. If you're enjoying Arcanist/Summoner, know that you can also gain access to the healer job Scholar that levels up at the same time as those two, so swapping over to healer should be pretty easy.