For example, Headshot simply increases Critical Hit damage. Do I get this bonus on my melee attacks? What about the other skills in the sniper tree? In other places, it specifies melee or ranged attacks.
I'm not 100%, but I think it should. If it states it definitively in other places, then it should be a global thing, despite the name. In fact, the Borderlands Wiki seems to agree: it simply states that critical hit damage increases by 4%. Not melee, not ranged, but just critical hits. In fact, considering that Headsh0t is used in an optimal melee build, it would seem to be correct.