Erm with regards to the cats... just don't. They should be OK in the same household but one should not be in the room when the other is loose. Its just to dangerous. Cat's saliva is also toxic to birds so if they are licking the cage bars or anything this is also extremely dangerous. Not to mention the antibodies your cat has- even though its probably healthy its still harmful.
I would never ever transport birds so i can't really comment on that. Its an extremely stressful process for them. I don't mean to judge honestly but poor birds
