So is it an alternative to: "Lew to kot'"? Can I say as well: "Tygrys to kot" and vice versa?
It is the same word, but used in different sentence.
Kot - is nominative- it is used when cat is a subject-Cat eats-=Kot je, and in sentences with to
Kotem is instrumental- it is used after certain prepositions, when indicating what tool is used, and after certain verbs. At the beginning you mostly use it after być=to be (jestem/jesteś/jest ...)