Mounted skirmisher + Improved Feint/Greater Feint? Yes? No? Why?
Yes, but Improved Feint takes your move action, leaving you with only a standard to do anything with. Technically they work together, but you're still using up your move action to feint. Mounted Skirmisher doesn't let you override that, it intends that rather than using up your move action normally to move with your mount you instead retain this action, which would let you full attack.
With this combination your mount could move to the enemy, you could feint as a move action, and then use a standard action to make a single attack. Without boht of these you would have only been able to move to the enemy and attack or move to the enemy and feint.