The most important things (apart from Pet Passport) are the final (24-48 hour) check with treatments and the size of the container.
She forgot to mention the most important timely Rabies vaccinations and following blood tests and the necessity for the dog to be microchipped. (She aint all that clued up on anything 'Cyprus'). Also you must use an appropriate airline approved pet carrier crate or kennel.
But then Oracle probably gets dispensation for that for her dog when she tells them that its owner is GREEK and the GREEKS invented the rabies vaccination and the microchip. She says she flies hers in and out regularly so SHOULD be able to advise but then she 'says' a lot of things!
DEFRA in Cyprus can give you full up to date info on what is required if you phone them.
They stopped the carrying of dogs as excess baggage a few years back and they had to go 'cargo'. Fare then dependent on weight of dog.
If they've started taking them as excess baggage again I haven't heard about it.
If your dog's rabies jabs are not up to date it will the process will need to be started from the beginning, and take some time before you can fly them. Approx 6 months I believe.