...so if this is the customary and time honored Vulcan greeting, how come it isn't always with the same hand by every Vulcan like a handshake or military salute?
Probably because A. Not all Vulcans have the same dominant hand, and B. Not all human actors have the same dominant hand.
I'm right-handed, and have no trouble doing the Vulcan salute with either hand.
As for hand-shaking... I dislike shaking hands, but when I do it, it's with my right hand. It was very awkward on one occasion when somebody tried to shake hands with me and he used his left hand.