I even did a Google search using that phrase against "Co słuchasz"
co słuchasz is colloquial and incorrect.
As the verb "słuchać" must be combined with a noun in the Genitive case, how can one say, "I am listening to ..." properly when many song titles don't always allow this easily?
Czego słuchasz? "Orała cień" Various Manx (song title in nominative)
or
Czego słuchasz? "Żółtych kalendarzy" Szczepanika (song title in genitive)
I think that the rule is that if it's easy to create Gen case then you do it but when it's not (
Czego słuchasz? Orła cieni would sound akwardly, even silly) then stay with Nom case.
To avoid grammatical troubles you can always use this expresssion:
Słucham piosenki o tytule (I listen to a song which title is...)
I would like to know how this can be properly used in other verb constructions which require the use of the Genitive case, np. "szukać".
The same applies here. Szukam piosenki o tytule...
Kind regards
You're welcome :)
PS
Ja uczę się języka polskiego, ale mówię trochę po polsku.
It's not a mistake but it doesn't sound natural when someone uses personal pronoun in such sentence. The verb form shows the person so usage of personal pronouns is required only in some contexts (to emphasize, for some stylistic purposes, when person form in conversation becomes unclear, etc). It would sound better to write just:
Uczę się języka polskiego, ale mówię tylko trochę po polsku.:)