Make sure to get a vet to certify the dog has its rabies shot. I've seen other pets on Egyptair, so at least it's possible. One worry would be the summer heat (depending on the date of travel). CAI is defintely NOT pet friendly, and I'd be very worried if my pet were out of sight in transit.

I like bhomburg advice to investigate Turkish or Qatar, as they will have better standards dealing with animals. Whatever you do, coordinate closely with the airline at every step, making sure someone is watching your animal at the airport, ensuring its secure in the cabin safety ect...

I've flown with pets several times here in Egypt, but always with KLM/Air France...which of course, do not fly to Ethiopia.

Good luck!