Advance after a fire combat
Infantry units in open order may shoot at enemy units in different hexes. How may this unit advance after a combat if different hexes become vacant?
It is not stated in the rules. I would suggest that after a unit in open order has fired and the target hex get empty, player decides if the unit advances (if with attack order) and then becomes disordered or if the unit fires at a second target. If it fires at a second target and the target hex get empty, the player decides if the unit advances (again, if aloud), then becoming disordered.
Advance after melee combat
If attacked hex becomes vacant, a unit must advance. But infantry in open order is not disordered after this advance (as after the advance in a fire combat). Is that correct?
It is disordered. See page 36, paragraph "After mêlée, if an infantry advances into a hex that is not open terrain..."
Infantry units in line may not move. Why are they allowed to do so during an advance after combat and a retreat? Is that prohibited only for normal movement?
I don't know why, but there is some kind of an answer from the designer somewhere in this forum (in a topic from KlausFritsch, if I remember***). Anyway, rules state that infantry in line cannot move during move action, but they can (must, after mêlée and with attack order) advance after freeing an hex, becoming disordered when they do so.
What about a unit in square? Same as with unit in line? May it retreat normally (not allowed to move) or does it lose a figure instead and do not retreat?
It is not in the rules, but, as Sergio said: no retreat, but loss of a figure. (I think that was clarified by the designers in some forum.)
Artillery reacting to a cavalry charge
If an artillery unit becomes charged by more than one cavalry unit frontally, in the same phase and from the same cavalry unit group at the same time, may it fire against all the cavalry units once and if so, does it have to do a morale test every time or only once?
If an artillery is charged by multiple cavalry units, only one cavalry unit is the attacking unit, the other(s) supporting the charge, giving a +2 modifier each (per "Charging with multiple cavalry units", page 38). Although it is not stated clearly, I would say that the artillery reaction fire is only against that attacking cavalry.
*** Yep! Right here!