I'm not sure about other forums. I've just been using this one, despite its ridiculously restrictive posting rules.
The limitations on number of posts and threads don't apply here in the New Player Help section. And to get around the 1000 character limit, post the first part of your message and then edit it to fill in the rest. (It's a good idea to have the whole thing written out in advance, so you can copy it and just click Edit Post and paste it in. That way you don't leave the introductory stub of a post you put in initially hanging there for long.)
Not really on quests. When you reach dungeon content, everyone entering the dungeon gets level-synced to within a couple levels of the level which that particular dungeon was designed for, so you'll be able to work together more effectively there. You reach the first full dungeons at level 15. There's also some quick mini-instances called guildhests that start at level 10, and those level-sync everyone the same way.
You can get a mount once you've joined one of the three Grand Companies, which happens at level 20 in the main storyline. They're not much faster than sprinting, but at least you can keep up the speed continuously as long as you're mounted rather than just for 20 seconds at a time.
Arcanist is a pet class and ranged debuff-oriented damage dealer (in that most of its attacks cause Damage over Time rather than a single burst of damage). One of it's advanced jobs, Scholar, is then the more buff oriented of the two healer jobs. (White Mages are a bit more reactionary, healing damage after it occurs, whereas Scholars have more skills for preventing damage.) It misses on one of your criteria, however, as it's a magic class.
The other one to consider is Archer. It's a ranged weapon, non-magic, damage dealer class with some support functions as well, at least once you advance to its associated job, Bard, which gets songs that can help your party members regain MP or TP. It's not, however, a pet control class.
In the early levels, you're going to be outleveling and so replacing equipment faster than it degrades, so fixing it isn't much of an issue. Once you gain a few levels and are holding on to gear longer, you'll want to keep an eye on its condition and repair items that are getting close to breaking, but it's pretty cheap and easy, and not something that you need to do very often.
There are three Grand Companies, one associated with each of the three city-states of Eorzea, and you'll join one of them when you get to level 20 in the storyline. They're allies of each other, though, and you'll be assisting all of them regardless of which one you join. There's little difference to which you choose besides the color of barding your chocobo will wear. (And once you reach max rank, you'll have the option of changing if you want.)
Most cutscenes can be skipped. (It's the circle button on PS3/PS4 controllers. I'm not certain on keyboard, but Esc is a good bet.) There are a few that can't be skipped, however, and that long intro cutscene may be one of them. It has one point where you have to answer a question, and your answer will determine one piece of your starting equipment (a ring the old peddler you were traveling with gives you when you arrive). You can't skip that part, so if it's implemented as one really long scene rather than multiple scenes one after the other, then you wouldn't be able to skip any of it. (I'm not entirely certain on whether you can skip any of the rest of it, since I've never tried, but I know you can't skip the part that needs your dialog selection.)