Yes, you definitely can since you are only the derivative asylee and not the principal applicant. I would also urge you, however, to apply for a Refugee Travel Document.
Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
Your post raises several issues. Some of those where addressed by my colleagues in previous posts. Other issue would be whether you risk to loose your passport if you "avail" yourself of the protections of the country from which you got a refugee status. Even a derivative refugee status. Well. You do run this risk. Whether to take the risk is up to you, but it is out there.
The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter,not should it be viewed as establishing an attorney client relationship of any kind.
Yes, you can renew at the consulate in NY. If you have had your GC for 5 years, I suggest you look into Naturalizing so you don't have to worry about maintaining your US legal permanent residency.
Best of luck.
This is not legal advice and is only general information. I am not your attorney. No attorney-client relationship is formed by this communication.