I have a battle in which an axis stack has to attack three adjacent hexes, one with a target marker and two with counterblow markers. Am I correct to assume that the defender can choose the defending terrain to be that in the target hex and get the corresponding defence shifts, but if the terrain in one of the counterblow hexes is chosen then there is no defence shift?
Yes this is correct.
Also, can the axis player apply a blitz marker to the entire attack or would it only apply to the single defending hex in which it is placed?
The Blitz marker applies to the entire attack.
This next question is really just for confirmation, as it's unambiguous in the rules but just seems a bit odd at first glance. In the supply phase of each player turn, it's always the initiative player who must check first, not the phasing player. I wondered why this is the case?
This is correct.
It is to keep some weird game situations from happening if one player passes and the other player continues the turn with his own offensive. Also, it puts the pressure on the player that has initiated the offensive, as if he passes and the non-initiative player moves first, he will beat a bit more of a disadvantage for having "yielded" the offensive to the other side.