Yes, the published errata has three separate cut and paste changes for page 117 of the Rulebook and one cut and paste change for page 269.

No. It is stated that they can move normally, but not that they can shoot. (Even being allowed to move and shoot normally when they use their "free" reform, the act of rallying disallows them from shooting).

Yes. Per the errata in Warhammer Chronicles they may shoot even when marching or reforming.

No. Per the rulebook, "...for charging, stand & shoot reactions, etc, the model needs to be facing the enemy as normal."

No. If a model in a Fast Cavalry unit can not draw line of sight from its base to its target without going through a friendly model, it may not fire.

Only if the character himself has the Fast Cavalry rule. Also note that characters without the Fast Cavalry rule may not shoot if the unit marches.

No. As specified in the Fast Cavalry rules, the improved line of sight only applies to models with the Fast Cavalry rule and only applies to shooting.

Per the errata in Warhammer Chronicles, an unlimited number of reforms may be performed, though no model may move more than its maximum movement distance.

No. This was an error printed in the Special Rules Summary Appendix and subsequently corrected in the Warhammer Chronicles errata.

No. The ability to move is limited to when the unit rallies on its "NEXT turn".

No. Per the errata in Warhammer Chronicles, any character model may join a fast cavalry unit and move with the unit, but they do not benefit from any of the special rules. However, please note that if you join a character on foot to a unit of fast cavalry, the character may slow the unit down as all units are restricted to moving at the pace of the slowest member of the unit.