Q:

On basic idea of a random variable.

Should one think of a random variable as a function, whos range happen to be subject to randomness? Or does something get lost thinking like this?
Could has been changed to should.

A:

IMHO you should exactly not think of a random variable as function, but as quantity whose value you are not sure about, which is why you describe this quantity via probability densities, expectation, etc.
This is despite the fact that formally random variables are of course defined as functions from a probability space to the real numbers, which is an admittedly ingenious construction. However, most often you are not interested in the probability space by itself, neither in the exact function, but really just in derived intuitive properties of the function viewed as random variable, like its expectation (this has an intuitive meaning independent of the underlying probability space. Only the formalization goes via integral of a function on the probability space), higher moments, and most importantly the probability distribution on $\mathbb{R}$ itself. In practice, sets of random variables are often equivalent when all these properties are the same, i.e., it is irrelevant with which probability spaces and functions thereon they are defined.

