The white rat the Rat King carries around with him is a homage to the horror film Willard (1971), where the protagonist is able to control rats and has a special affinity to a white rat with the name of a Greek philosopher (Willard's rat was named Socrates, while the Rat King's rat is named Aristotle).
According to Kevin Eastman, the stray cat is meant to be Klunk, a stray cat Michelangelo found and adopted in the original Eastman-Laird TMNT comics.
The Rat King has named his personal white rat Aristotle after a Greek philosopher, and named his rat subjects after Roman figures (Caligula, Claudius, Nero and Commodus were Roman rulers).
The movie theater that the giant rats destroy is playing a movie called food of the vermin. This is a reference to the movie Food Of The Gods which is about giant rats who grow large because of mutated food and attacks people.