Hip Shooting and Sprint
Yes. Why wouldn't they?
Because the Sprint talent states when taking a Full Move Action, in capital letters, thus referring to the actual action.
And Hip Shooting is a full action, which includes shooting and moving up to his Full Move rate.
By RAW, [Hip Shooting] is suggested to be its own Full Action that is basically a Full Move. Personally, I see no harm in letting [sprint] affect [Hip Shooting]. If it looks like a Full Move, feels like a Full Move, and acts like a Full Move, it's probably a Full Move. Otherwise, it's just a matter of semantics.
Edited by Asymptomatic
Personally, I'm completely with Asymptomatic in regards to not seeing any harm in letting it work, if only for sake of player sanity.
By RAW, however, I think it's fairly clear that Hip Shooting is it's own action that lets you move as a Full Action and take a shot, whereas Sprint let's you add your Agility Bonus to your movement when taking a Full Movement Action.
It's actually a nice difference, between the two, and allows Sprint to negate the devaluement of the Full Move Action that Hip Shooting would induce.