You want to include city within your airport dimension. You are intentionally flattening out your normalised schema to aid the speed of the dimensional model which can seem counter intuitive if you are coming from transactional development.
With regards to the perennial child relationship, you want the parented to be translated into the surrogate of the region record. Ssas will provide the functionality to relate parent child records when you are designing your cube.
Multiple facts are not unusual, but unless the fact data is completely unrelated, there is no need to separate them into different cubes. The requirement for multiple facts will be driven by having data at a different grain. Keep all of you metrics (I.e. Flights) together, but you would separate out flight metrics from food sale metrics