Ludzie mają zwierzęta ... ale koty mają ludzi.
Could I use both, ludzie and mężczyźni, when referring to men?
Yes. This course will probably require you to use specific terms, but in real life it works just as it does in English – "ludzie" are "people" and "mężczyźni" are "men".
I can also say that both ludzie and kobiety refer to women.
I didn't understand why the ending is "zwierzęta". Could someone help me please?
Neuter nouns with the nominative ending -ę are quite rare and undergo irregular declension: