Do I have legal recourse for a severe allergy provoked by neighbor allowing her cat to play in the hallway just outside my apartment?

You could sue the neighbor for negligence and may also have a claim for negligence against the landlord.  The case against the landlord may be rather weak but you could claim the landlord was negligent for not taking corrective action to prevent the neighbor from playing with her cat in front of your apartment if the landlord had notice of your allergy to cats. There would only be one lawsuit naming both your neighbor and the landlord as defendants.

Negligence is based on the failure to exercise due care to prevent foreseeable harm.  Due care is that degree of care that a reasonable person would exercise under the same or similar circumstances to prevent foreseeable harm.  In order to prove negligence, you would need to prove breach of the duty of due care, actual cause, and proximate cause. 

Breach of the duty of due care occurred here when the neighbor refused to play with her cat away from your apartment despite being aware of your allergy.  Actual cause means but for the neighbor playing with the cat outside your apartment would you have had an allergic reaction?  If the answer is no, you have established actual cause.  Proximate cause means were there any unforeseeable intervening events that would have relieved the neighbor of liability?  If the answer is no, you have established proximate cause. 

Your damages (the amount of compensation you are seeking in your lawsuit) would be your medical bills, compensation for pain and suffering and documentation of any wage loss.  You would need to obtain your medical bills and medical reports as well as documentation of any wage loss if you have an allergic reaction due to the neighbor's cat being outside your apartment.  The medical reports will document the nature and extent of your injuries and will be used to determine the amount of compensation you receive for pain and suffering.  Compensation for pain and suffering is an amount in addition to your medical bills.  Compensation for the medical bills and any wage loss is straight reimbursement.  If you have an allergic reaction to your neighbor's cat being outside your door, you will need to file your lawsuit for negligence prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.  The landlord could assert the defense of assumption of the risk to your lawsuit for negligence.  Assumption of the risk means that you recognized and understood the danger and voluntarily chose to encounter it.  The landlord will claim that you knew that pets including cats were allowed on the premises when you moved into the apartment complex and therefore you recognized and understood the danger of an allergic reaction, but moved into the apartment anyway.

Apart from the above discussion of negligence, you may have a legal argument against the landlord for breach of the covenant of quiet enjoyment.  In every lease, there is a covenant of quiet enjoyment which means that the tenant cannot be disturbed in his/ her use and enjoyment of the premises.  If the landlord has been notified of your allergy and the neighbor's refusal to play with the cat elsewhere, and the landlord has not remedied the situation within a reasonable time, you may have an argument for breach of the covenant of quiet enjoyment.  This may be a weak argument because the landlord could claim that you knew that pets including cats were allowed when you moved into the apartment complex.