Can a person develop lactose intolerance?

Yes, comes with age. The great majority of the world's population becomes lactose intolerant with age (except northern europeans), because people produce less and less lactase enzyme and will not be able to digest milk sugar. Lactose intolerance can come on more suddenly after diarrhea. Taking lactase (lactaid tablets) can help a lot. We humans are the only mammals who keep drinking milk after weaning off the breast.
Absolutely. Lactose intolerance is very common. It is the loss of the lactase enzyme. Humans are the only animal that maintain the lactase enzyme into adulthood, but 1/3 people lose this enzyme.