Should "You get the cats of mine" be correct too for the translation?
No. It won't be a correct translation.
Here, the meaning of [van mij] is [from me] .
Check the translation of "van" in the dictionary. It also means (from).
Sinve "Van mij" can mean both from me and mine, how would you say "of mine", as in "Take a piece of mine"? Would you say "van mijn" or would you still just say "van mij"?
Take a piece of mine: neem een stuk van mij. But when you put a word behind it, like cake, it changes. Take a piece of my cake: neem een stuk van mijn taart.
Je is also accepted. Though if it is a type what you hear exercise only jij is accepted here due to the difference in pronunciation.
In english we can take something "from" someone or take something "off" someone. Does van.. account for both of these or is their a difference in dutch too?