"Do you have a cat?"
Translation:У тебя есть кошка?
In this case, there being a cat or not is the question. "Do you have a cat?" We question the existence of a cat belonging to the person we address with this question. The verb есть cannot be omitted in that case.
If the existence of the cat is not the question, but simply the location ("Is the cat at your place/with you?") , then "у тебя кошка?" is acceptable and correct use of the language.