Loot:
As erikdaae and Bingopajamas mentioned, dungeon loot is handled by Need/Greed/Pass for significant items and a random party member for minor ones. When you're on the open landscape and fighting monsters for drop items, it's a little different because you won't necessarily be grouped with other people who fight the same mobs. In that case, there's two ways of earning loot.

1) If you're the first person to strike a mob, you're guaranteed any loot it drops as well as XP and kill credit for quests, hunting log, etc. (Mobs you tagged this way are identifiable by a red health bar, and those tagged by other people by a purple one.)
2) If you're not the first person to strike it, but you do at least 1/3 of the damage to kill it, you also get full loot and kill credit (so mobs can end up dropping double or triple their normal amount of loot if multiple players are involved in killing them).

So helping never decreases the loot the first person who tagged it gets, even if the helper ends up getting loot as well.

Mount
The mount doesn't strictly come from the main quest line, but the main quest line will have you join a Grand Company. Once you do, there's a quest to get a Chocobo, but you'll have to earn a bunch of GC seals to do so (2000 of them IIRC). The chocobo quest itself is actually identified as a side quest rather than a main storyline one (presumably because they didn't want to stop the story there long enough for you to earn that many seals), but since it has a main storyline prerequisite and pretty much everyone starts working on the chocobo quest as soon as the main storyline unlocks it, people think of it as coming from the main storyline.