I want to write a character (in the Percy Jackson Universe) that has extreme trouble falling asleep. However, they are a child of Hypnos, God of sleep. What illnesses would prevent them from sleeping? Insomnia or Nightmare disorder maybe? And (if you want), how would his sleep problems interact with the fact that their dad is the god of sleep?
Major Depressive Disorder has insomnia built into the criteria. Insomnia is also a possible symptom of a manic or hypomanic episode, as part of a Bipolar I or II diagnosis.
While not in the criteria, some anxiety disorders may prevent your character from sleeping because of their constant worrying. OCD could also work if your character is driven to perform time-consuming compulsions before they can go to sleep.
There is also a prion disease called fatal familial insomnia that causes insomnia; however, the lack of sleep caused by this disease is universally fatal, usually within one year after beginning to show symptoms.
In addition to insomnia or nightmare disorder, I have a bit of an unorthodox thing to suggest: central sleep apnea. Basically, the character’s brain forgets to tell their body to breathe while they’re asleep. While it doesn’t initially prevent someone from getting to sleep, it does mean that the character would be waking up frequently and not getting restful sleep. This may or may not work with your story, but I thought it’s a neat, unusual idea.
As for your last question, honestly, I have no idea. However, a certain proverb comes to mind: “The shoemaker’s children always go barefoot.”
Hope that helps!