Well actually, I suspect that it isn't arbitrary at all: historically, people have considered left-handed as deviant, strange, sinister, so obviously the good molecules are right-handed.
(Disclaimer: I'm a slightly bitter left-handed designer who has seen one too many "more intuitive" designs that are optimised for right-handedpeople)
The "left" or "right" refer to the direction that the chemical (usually in solution) rotates the plane of polarized light passing through it, from the point of view of an observer that the light is traveling towards.
An experimental setup to demonstrate this is a light source, a linear polarizer oriented in some direction (say along the x-axis for definiteness), followed by the sample container, followed by another linear polarizer, followed by a detector.
With nothing in the sample container, the second polarizer is rotated around the light beam axis until no light passes through. This will be very close to a right angle with the first one (try it with polarizing sunglasses :) ). The sample is then added, and the change in angle of the second polarizer required to again allow no light through is observed.
I don't follow your question. A right-handed helix remains so regardless of its orientation, just like how I wouldn't become left-handed if I were suspended upside-down.
That's the definition of chirality. If you have a tetrahedral orientation of four different atoms around a center atom (e.g. carbon), there will exist a "right handed" and "left handed" version.
You can rotate the "left handed" version any way you want, but you won't be able to superimposed it on the "right handed" version.