That is true, but the sentence "I don't have chickens" sounds strange. People would use the verb "own" when referring to the animal, and "have" with the meat. At least, in my experience, since people, for the most part, tend to eat chicken and not other animals like dogs or cats.
This is a comment on English usage not Italian. But- I believe that using "haven't" to indicate not having something is more common in the U.K. than in Canada and the U.S. In North America most people say "I don't have any cash" rather than "I haven't any cash." Both are completely correct.
Indeed. To have can be used as a regular verb (=to own, to possess) or as an auxiliary verb. As a verb, the negative is formed using the "don't/doesn't"; as an auxiliary "hasn't/haven't".
The same form goes for question:
Do you have a dog? (verb: possession. Not *have you a dog?) Have you met him? (auxiliary).
Note: in "have got", "have" is a verb": "have you got a car?" "No, i haven't got a car" :-)