Why is Stay Puft even in the cartoon?

That was never explained in the cartoon. One possibility is that after the ecto marshmallow was cleaned up, it was stored in the Ecto Containment Unit and recongealed in the same form it last had. But since Gozer's intelligence had been shifted back to it's own dimension, Stay Puft was mindless and then developed a simple, childlike personality. The "real" reason is like Slimer, he was very popular with kids from the movie so it was only natural to bring him in in some way to capitalize on his popularity.