Making sense of measure-theoretic definition of random variable

But, what should be $(Ω,F,P)$?

Depends on your application. If want to look at arbitrary random variables, then it is simple arbitrary. If you have a specific example in mind, it is not.

How does this relate to the elementary computation [...]

In this case you assumed that $X \sim \mathcal N[0,1]$. In particular $X$ is a continuous variable. In measure theory we say that the distribution of $X$ is absolutely continuous w.r.t. to Lebesgue measure. The Radon-Nikodym theorem then guarentees the existence of an $f_X$ with the property you have stated, so that we can apply the change of variable formula to make the computation of the expectation easier. Without the change of variable formula, we would have to compute the expectation with the definitions of expectations for indicator functions then simple functions then (?simple integrable then $L_1$ then?) nonnegative functions and then measurable functions.. But, again, this is a particular example, where $X$ is continuous. The measure-theoretic definition of expectation is much more general.

What is the meaning of $P(dω)$?

It doesn't mean anything. It is a notational crutch, like writing $\lim\limits_{n\to \infty} a_n$ instead of $\lim a$. It becomes useful, if you have multiple nested integrals / integrate w.r.t. to product measures.