No, the modifier will still apply. For example, if a unit loses a round of combat to a Terror causing enemy, the -1 modifier to Leadership caused by Terror applies to whichever Leadership characteristic the unit makes its Break test against, be that its own or the General's.

Not whilst that character's unit is still engaged with the model that issued the challenge. Battle standards are somewhat less inspirational whilst being used as a blanket to hide beneath.

No. The points you pay for a chariot include its crew, whether that chariot is included in your army on its own or as a mount for a character. However, in reality it might prove difficult to fit a character and the full complement of crew into the chariot model itself. If so, it is perfectly acceptable to remove one or more of the crew models to make space for the character. In such cases, we can think of the missing crew model's profile(s) as a way of representing that the beasts pulling a character's chariot are bound to be bigger, stronger and better trained than normal.

No. You must use a magic item fully or not at all.

No, you choose whichever armour value is better. However, allowing for magical armour and other special rules, 'better' is somewhat subjective. Obviously an armour value of 4+ is better than one of 6+ but there is often more to deciding which armour to use than just the armour value it offers, many types of magic armour confer additional bonuses that make them better. Basically, when the rules say 'choose', they are giving you a choice.

No, a shield is carried by the character, not the mount.

No.

No. Characters can move through the ranks into the fighting rank, but cannot move within the fighting rank.

Yes.

Yes.

Yes. The rule that a lone character cannot be shot at unless they are the closest target only extends to units they are within 3" of. If a unit is more than 3" from the character, it does not protect them, regardless of how close it is to the enemy.

No.

The purpose of the Targeting Lone Characters rule is to protect characters from enemy shooting, even from shooting that does not follow the usual rules (though it does not protect them from templates). Therefore, and in the spirit of this rule, unless they are the closest target to the cannon, an enemy character that is within 3" of a friendly unit that contains five or more models of the same troop type cannot be struck by a cannonball, even if their base lies directly under the path of the bouncing cannonball. However, if a Lone character's base lies directly underneath the strike point of a cannonball, they are hit.

No.

They must use their magic weapon. However, some named characters will carry a 'back-up' weapon which they can use should their magic weapon be destroyed.