No. As stated on page 143, models that know such spells can attempt to cast them when chosen during the active player's Shooting phase. Perhaps obviously, this includes Bound spells.

No. A model cannot shoot during the Shooting phase if it marched during the preceding Movement phase and, although they work differently to most missile weapons, Breath weapons are a type of missile weapon.

Yes.

No. To claim the benefits of being on a hill, a unit must be entirely on the hill.

A unit that is closer to the top of a hill (i.e., the centre of the hill or the edge of the battlefield) can draw a line of sight across and shoot over one that is closer to the bottom. Units cannot draw a line of sight across or shoot over units that are closer to the top of a hill than they are.

Missile weapons mounted on chariots (or howdahs) are shot by the crew, using their Ballistic Skill.