This game has some weird things.

[Kalsam]

You can edit a post to make it longer, just as an FYI

I'm just gonna run through each, just cause it'll be easier to give my thoughts on each bullet point in order.

Damage Numbers - Never really bothered me personally. But then, I'm rarely paying attention to such quick flyby things, that even in other games, with mods I'll move the flying numbers way off to the sides. lol

Chocobo Names - Hadn't actually noticed this one until you mentioned it. After years of playing games where one colour meant Not-PvP-Flagged and another meant PvP-Flagged... I just kinda ignored it.

Combat Music - Can't comment, as I turn off all the music in game anyway. After listening to it the first couple weeks, I got bored of it and just play my own music.

FATE Level - This is more due to the way they want to have you progress. Every level below a FATE will reduce the Experience by something like 2-3%. This is to encourage you to find FATEs in your level range. Kind of annoying, but I can see the merits to keep every area with FATEs a bit more active.

Instant Spells - Oy vey... This one is a huge complaint in the tanking community. Specifically for our 'invincible' Cooldowns. Would be really nice if they went off at the beginning, but still played the animation. >.O

Melee Classes Animation - This actually happens in every game with animations for character attacks. (Which is... all of them?) It's just less noticeable in most others because we don't move as much.
This is because when the developers are designing the attack animations, the character is set to a 'ready' pose, animated and returned to the 'ready' pose. If you actually jump nearly constantly while fighting on a melee class in FFXIV, you don't see the 'slide' as often cause you keep interupting the attack animation with a jump animation. (A few still lock in, giving the attack animation, but it's less than if you're not jumping. In fact, one some classes, you can actually get some pretty sweet looking 'moves' by jumping when you're attacking on the move. lol)

Flying - This happens in a fair amount of games. FFXIV just has a slightly higher range in the Z axis for mobs to attack in due to some combats with motion/movement. Just fly a little higher.

Solo!MMO - They went too much with the idea of a singular hero (ie: you) for the story. Meaning that to put others into the storyline, you'd suddenly lose your status as the hero in the story. Very few games have managed to actually make the idea of 'You the Hero' work, without breaking the storyline. =\ Even WoW's had to go to a more solo route lately (Garrisons) due to how important your character's now become in the story. Kinda hard to be a champion when everyone else is one too. :P

Tab Targetting - I agree, the prioritizing on this needs to be switched to 'closest to character' over 'closest to camera'. Currently the only way to get around this atm is to move your camera with Free Look to a position that forces the tabbing in the method you want. (ie: Pull, run through and make sure your field of view only has what you want to fight. Or use a top-down camera.)
Edit: And the filtering options Dhex mentioned, though... as I also just mentioned, this won't always work properly in some rare cases >.O