A. Under charge reactions, when you flee, you immediately turn the model/unit about its center so it is facing directly away from the center of the enemy unit and then move in that dirction.
B. The rest of his unit just stands there -- kind of like in the scene from the movie Troy where Patroclus and Hector duel in the middle of the battle (and just an fyi, you don't actually physically step aside outside of the unit). The rest of the unit stands there until the challenge is concluded. The only thing that can contribute to the challenge is a mount being ridden by the participants. And yes, you could charge the unit with one of your own and fight as normal.
C. Yes, you would calculate any wounds caused in the challenge and then all the other factors -- how many ranks, banners, charging, etc.
Lone Characters cannot pivot at will -- they move during their Movement Phase where they can pivot to their heart's content -- except when charging as they still have to wheel and only get one). Lone Characters are treated as units of the appropriate troop type for all rules purposes, so yes, they have arcs just like units, so they can be charged in the flank and rear.