The Birmingham Zoo is a private non-profit 501c3. Private property laws apply. CC isn't illegal, the facilities are open to the public. Any signs don't have the force of law, but you may be trespassed if you carry and someone notices.
That part of town isn't noisy by any stretch. Homewood is almost a police state, Mountain Brook is very law-abiding.
Unless you're in Birmingham proper (in the city itself, in the bowl north of Red Mountain, around Five Points and other places), crime is generally low.