1. No idea.
2. Yep.

1. What does Agility affect? When a gun grants +2 agility... what am I getting? More speed? More jump height? Higher armor class rating?

Agility affects non-sprinting movement speed and jump height.

2. I'm slowly losing Special ammo after each death in the Crucible. Is this normal and I never noticed? Or is it because I'm using Invective without having the ammo regen perk yet? It's hard to keep track of though... I swear sometimes I gain a shot but mostly I seem to always be really low on Special ammo without firing my gun shotgun...

You will lose ammo for your Special per death. I don't know how much, and I think it has a minimum amount, but I don't know the full details.

1. What does Agility affect? When a gun grants +2 agility... what am I getting? More speed? More jump height?

I think: non-sprint max movement speed, movement acceleration, jump height.

You feel lighter on your feet, snappier.

Higher armor class rating?

Armour is different stat. Put your cursor over your class icon in the character screen. You'll have Armour, Recovery and Agility. It affects that agility stat. But it won't show up on this screen. You'll only feel it.