I think that using your miles for a low season roundtrip on AA to enither ATH os IST, and then buying a ticket for your onward journey to Cairo is an excellent plan. Not only is it a smart use of your miles, it gives you an opportunity to visit another country or two.

Personally, I would recommend that you make IST your first destination and that you consider a stopover somewhere in Europe depending upon your carrier. There are great connections between IST and CAI.

I would forget Cyprus (LCA) as the times of arrival and departure are usually at the most inconvenient of times and compared to (say) IST, it is rather a boring destination (not to mention expensive) for a short stopover.