Do international tournaments allow players to participate in more than one event?
For example, lets say Lin Dan wishes to participate in MS, MD and XD and China allows him. He'll likely be too tired playing three matches in a day and won't win anything but do the organizers allow/accommodate for this?
Yes, they do. In fact, it's quite common for the top doubles players to also play Mixed. For example, in the German Open, Kamilla Rytter-Juhl and Mads Pieler Kolding both won the Mixed and doubles competitions.
It's pretty rare for a player to compete in all three disciplines (and then be good at any of them), but even that is possible, and happens regularly in Junior tournaments. Note that Lin Dan is not ranked in Mixed or Doubles and thus would not qualify without intervention from the tournament organizers, although he could of course pick a partner that has shown great doubles results in the past.
Tournament organizers and the referee will schedule games so that players have some rest time after their games (typically 30 minutes between matches). In the first rounds of Junior tournaments, you'll often see energetic players waiving their rest time though, especially in the evening.