The BRB does in fact cover all your questions.
1. No. BRB p. 68 (Fast Cavalry):
Free reform: Unless it charges, a Fast Cavalry unit is allowed to reform as many times as it wishes during its move, even if it marches, provided that no model ends up moving a number of inches higher than double its Movement value. [Italics mine]
Note that you cannot charge after a Swift reform either, because that takes place during the Remaining Moves sub-phase (BRB p. 95).
2.
In the case of normal movement:
BRB p. 14: "When a unit performs a wheel, every model counts as having moved as far as the outside model. Once the wheel is complete, you may use any movement that the unit has remaining. A unit that is not charging can wheel several times during its move, and indeed can mix forward movement and wheeling, as long as it has enough movement to do so. Units are not allowed to wheel backwards."
In the case of a charge (BRB p. 20), "the unit must move straight ahead, except that, during this move, it may make a single wheel of up to 90°." This wheel is free, as a "charging unit can move an unlimited amount — it's already been found to be within charge range, so we don't worry about distance from this point on" (loc. cit).
3. Charging units can make supporting attacks as normal. However, in the case of units that have the Fight in Extra ranks special rule, "supporting attacks can be made by an extra rank than normal on a turn in which the unit did not charge" (BRB p.69). [Italics mine]
BRB p. 82: "When a cavalry model makes supporting attacks, only the rider can strike — we assume that the mount lacks both the discipline and the reach to fight through those ahead of it." This applies to Monstrous Cavalry too.
4. BRB p. 95: "If only the standard bearer and/or the unit champion remain beside the musician then the musician must be the next casualty."
Keep in mind that a champion can be slain earlier, if he is specifically targeted.
5. BRB p. 35: "If the dispel result equals or exceeds the spell's casting result, the dispel is successful and the spell does not take effect."
However, in the case of RiP spells (BRB p. 36): "a Wizard does not need to beat the original casting dice roll if attempting a dispel in subsequent turns, but rather the minimum casting value listed in the spell's description (so there is no need to make note of each spell's casting roll). For this reason, remains in play spells cast with irresistible force can be dispelled as normal in subsequent turns."
6. BRB p. 82: "Cavalry have the Swiftstride rule." This applies also to Monstrous cavalry and is not affected by barding (as that is not specified by the BRB).
BRB p.76: "When fleeing and pursuing, units entirely made of models with the Swiftstride special rule roll 3D6 and discard the lowest result."
BRB p. 57: "A unit that does not pursue can perform a reform manoeuvre." You can either attempt to restrain or pursue/overrun.
During pursuit, "you move straight forward a number of inches equal to the amount rolled on the dice."
However (BRB p 58), "If a pursuit move would take the pursuer into contact with an enemy unit, then the pursuers must charge the enemy unit.
Carry out the charge as you would in the Movement phase, following all the normal restrictions. However, you do not need to roll for the charge range — we already know from the pursuit roll that these unwitting chargers have momentum to reach the foe, whether they wished to or not. The charging unit must wheel and close the door in such a way as to maximise contact, as they would with a normal charge."