Yes, as they have many parts that are wood or metal (i.e. unliving). Note that Undead Chariots are also immune to poison.

The pursued unit is destroyed as normal, and the Chariots will move into the woods and suffer hits.

Strictly, there is no rule to stop such behaviour, and in some cases it is unavoidable or appropriate. However, deliberately turning a Chariot so when it charges it clips the enemy's unit is bad form, and to be frowned upon. See page 266 of the Warhammer rulebook for a fuller discussion on the horrid creature that is 'clipping'. View online.

Yes. It's still in the Close Combat phase, just before anything else happens (with the notable exception that the resolution of wounds caused by warpstone vapors from Skaven Clan Pestilens plague censers is resolved before impact hits).

Yes.

Regardless of the challenge situation, a character in a Chariot is considered to have stepped down from the Chariot for a challenge and therefore does not benefit from the Chariot's enhancement to his armour save. In addition, regardless of the challenge situation, the Chariot's attacks are not lost. If the enemy character engaged in the challenge is in a unit made up of multiple models, all the chariot attacks (impact hits, crew, and beasts) strike the enemy unit. The enemy character can only attack the friendly character per the rules for challenges, and the enemy unit can attack the friendly Chariot. If the enemy character is a single model the friendly Chariot can attack the enemy character as if the Chariot was in the challenge, with one exception: when the enemy character is riding a monster, impact hits are applied against the monster, not the rider. In this situation the enemy character (and his monster) can strike either the friendly character or the friendly Chariot. Also, remember that creatures pulling the Chariot can only attack enemy directly in front of them.

This means that creatures pulling the Chariot can only attack enemy models in their front zone. This includes enemy models in corner to corner contact in their front zone. The creatures may not attack enemy models in their flank (or rear) zones due to the restrictions imposed by harness and reins.

No. In the opinion of the Direwolf FAQ Council the S5 hit only applies to situations where the character is considered to be in the chariot at the time the chariot is destroyed. As the character has lost his armour save bonus due to stepping down from the chariot and is considered to be fighting on foot at the time, applying the S5 hit for the character being thrown from the chariot does not appear appropriate in this circumstance.

Strictly by the rules, unless the Chariot is compelled by a psychology (e.g. frenzy, failed panic test, etc.) to move, then it is stuck for the remainder of the game. Page 127 of the Rulebook states "Chariots cannot voluntarily move over obstacles or difficult terrain...". Note however that it is the opinion of the Direwolf FAQ Council that this situation was not thought of during the writing of the text and the Council recommends that in this situation the owning player be allowed to move (or charge) out of the difficult terrain (provided the Chariot survives the additional D6 S6 hits it will immediately take for moving through difficult terrain again).

Yes, the rules state that you can place the character anywhere within 2" of the destroyed Chariot, and there was a legal charge declared against that enemy unit. The owning player could not place the character into combat with a unit within 2" that the charge was not declared against.