Can doctor / dental x-rays go through the X-ray machines at the airport without being damaged?

Yes if developed. If already processed and developed by the dental office, the x-rays should be fine.
Airports and xrays. As long as the x-rays films have already been processed and developed, this should not be a problem at all!
Yes. They should be fine, they should have been processed and fixed so it should be good, if they were digital then of course.
Yes. Once an x-ray is developed the image cannot be change by exposure to more x-rays.