Why is the genitive used here? I thought that was used to show possession.
Well, there are only 7 cases - so it would be hard to have one only for possession. It has numerous usages. Luckily, there's a brand new topic on that: check here.
What is the genitive plural of mężczyzna?
Look at tables of declension:
Would it be incorrect to say that "słuchać" means both "to hear" and "to listen." Which is the closer meaning?
It doesn't mean both. "słuchać" is to listen. "słyszeć" is to hear.
Słuchacie tego mężczyzny? - Are you listening to/ Do you listen to
this/that man?/ Are you obeying/ Do you obey this/that man?
Słyszycie tego mężczyznę? - Can you/ Do you hear this/that man?
True, it technically could mean "obey", added.