Is there a cat cat or a cat dog
No. There is a cartoon called "Cat-Dog". But, there is no cat-cat or cat-dog in reality.
wiki.answers.com/Q/Is_there_a_cat_cat_or_a_cat_dog

What if your cat is a black cat what type of cat is it
Unless you got the cat and pedigree papers from a breeder, it is almost impossible to tell
wiki.answers.com/Q/What_if_your_cat_is_a_black_cat_what_type_of_cat_is_it

Will other cats chase your cat into your cat door
Well, my cat has been chased in by our other cat. so it is possible for that to happen to yours.
wiki.answers.com/Q/Will_other_cats_chase_your_cat_into_your_cat_door

What has a head like a cat a tail like a cat and feet like a cat but is not a cat
racoon
wiki.answers.com/Q/What_has_a_head_like_a_cat_a_tail_like_a_cat_and_feet_like_a_cat_but_is_not_a_cat

Has a head like a cat feet like a cat tail like a cat but it is not a cat what is it
I'm guessing a raccoon.
wiki.answers.com/Q/Has_a_head_like_a_cat_feet_like_a_cat_tail_like_a_cat_but_it_is_not_a_cat_what_is_it