As a woman whose main character is male (for pretty much the same reason) I've never seen an issue with it. It seems to be pretty common, in fact.
The only thing is that you need to be aware that in a game where we don't know the players' genders, the usual default for choosing pronouns is to base them on your character's gender. If you're going to freak out whenever someone refers to you as "she" then maybe you shouldn't play a woman character. But if that doesn't bother you, then go ahead, enjoy your pretty girl character.
If you're going to micromanage your pet's actions (which is only really necessary in hard content) that takes some extra effort regardless of what control scheme you use. Personally, I think the controller is easier than keyboard, but it may take a bit of setup to get skills where you want them. (I recommend turning off the built-in "pet hotbar" and adding the pet skills to a regular cross hotbar —perhaps the one you use as your extended crossbar— so you can arrange them better.)
But I've heard Summoner and Scholar are among the jobs which are getting a pretty major revamp when the next expansion hits, with pet actions being treated more like character actions, so we'll have to see how it goes after that.
Not in the big, flashy, visually exciting sense. Summoner pets are thematically related to primals that we've defeated, and those primals are classic Final Fantasy summons. But the pet versions we create from their leftover aetherical energy are much smaller and less visually impressive (presumably because a full blown classic FF summon would be too distracting to the rest of the party, but there are also story reasons why summoning a full-blown primal would be a bad thing).
While there are some free companies looking specifically for hardcore players to join their raid parties, casual social FCs seem to be more common. So long as you're honest about what you're interested in when asking to join them it shouldn't be too hard to find a like-minded group.