For an arbitrary periodic function p(x), whose period and Fourier expansion might have been known in advance, how can we get the Fourier expansion/coefficients of |p(x)| from them?
Or, if possible, is there some good polynimial approximation for absolute function/step function?
No, you cannot get the coefficients of $|f|$ from the coefficients of $f$. Consider that $\sin x$ has a very simple Fourier series (just one nonzero coefficient), but $|\sin x|$ has an infinite series. You can't really get the latter from the former.
This applies generally to nonlinear transformations (other than composition with polynomials, which translates into convolution)
Concerning approximation of $|x|$ by polynomials: this is possible on finite intervals (see Sequence of functions that converge on absolute value) but not uniformly on the entire line. So, if $f$ is unbounded, approximating $|f|$ by $p_n(f)$ is very problematic.
Bernstein polynomials, mentioned in the above thread, give a constructive approximation to $|x|$ on a finite interval. Given the importance of this specific function, there is some research on getting a better approximation for it: see the introduction of this paper. But in practice, if you want a good low-degree polynomial approximation to $|x|$ on a given interval, use a numerical routine like this one.