Autocorrelation of filtered cyclostationary random process
Your first two equations generalize to the spectral correlation and cyclic autocorrelation. The most general results are (12) and (13) on my blog post on input-output relations for $n$th-order cyclic cumulants and $n$th-order cyclic polyspectra: signal processing operations and cyclostationary signals. 

You can find the generalization of your first equation in several places, including Professor Gardner's book "Statistical Spectral Analysis", Eq (90) in Chapter 11:

$$S_y^\alpha(f) = H(f+\alpha/2) H^*(f-\alpha/2) S_x^\alpha(f)$$

And yes, the output of the filter is also cyclostationary with the same period provided that the transfer function $H(f)$ is such that $H(f+\alpha/2)H^*(f-\alpha/2)$ is not zero for at least some $f$ for which $S_x^\alpha(f)$ is also not zero. In other words, the filter could remove so much of the energy of the signal that the remainder is no longer cyclostationary.