Do I require a refugee travel document as a legal permanent resident who was previously an asylee?

Although you may or may not not have a problem traveling to Armenia with your Egyptian passport, it would probably be best for you to apply for a reentry permit and travel on that instead. As a permanent resident, it is more appropriate for you to apply for a reentry permit than a refugee travel document. The reentry permit has pages on which visas of other countries can be stamped.Due to the limitations of the Lawyers.com Forums, Alan Lee, Esq.'s (the "Firm") participation in responding to questions posted herein does not constitute legal advice, nor legal representation of the person or entity posting a question. No Attorney/Client relationship is or shall be construed to be created hereby. The information provided herein by the Firm is general, and requires that the poster obtain specific legal advice from an attorney. The poster shall not rely upon the information provided herein as legal advice nor as the basis for making any decisions of legal consequence.  

If you are a lawful permanent resident, your green card (I-551) serves as your travel document (together with your passport).  Since your nationality hasn't changed, you won't have any trouble at the border when you return and show your Egyptian passport and your green card.