Is 너희 another word for "you" in singular?
It would be you plural usually.. It doesn't make sense in this sentence though.
너희 = you plural (but not commonly used), 저희 = 우리 formal (also not commonly used)
In Slavic languages, plural you (ви) is used in polite speech instead of singular you (ти). Could that be the case here as well?
Haha, not in every slavic language. In some third person is used in polite speech, like "could mister do something?" instead of "could you do something, mister?"
I thought it might be plural but as you say, it doesn't make a lot of sense. Thanks!
It's more like we, but including the listener. 우리 is a bit less common but excludes the listener
Why is it not you make fun of the cat?
Cat is the subject here indicated by the 가 at the end.
Because of the particles used after cat and you, you is the object and cat is the subject of the sentence
The cat is the subject (가) and yoi is the object (를)
Because you has the marker for which the verb is modifying
The subject always acts on the object, so since cat is marked with 가 (the subject marker) and "you" is marked with 를 (the object marker), the cat is going to be making fun of "you".
"You make fun of that cat" would be 너희가그고양이를놀려요.
"가" is the topic marker and "를" is the object marker
The cat is the subject (since it ends in 가) and you is the direct object (since it ends in 를)
이/가 is attached to the subject and 을/를 is attached to the noun of the verb