ETA: Based on your added info that he drinks virtually nothing, I'm not surprised at all that he's not getting over the bug or that he has headaches! You need to find other water-filled foods for him if he balks at water! Jello (the less dye, the better), popsicles, watermelon, berries, all-fruit juices, applesauce, clementines/mandarins, and anything else. Don't discuss it, don't nag him - just make it yummy and fun! You can get a good isotonic sports drink that has no artificial dyes and no fake sugar (just fructose) in it - that's what we all drink, and my son/husband are major runners. Find out why he doesn't want to pee - is he nursing a UTI? Does he have fears about the bathroom? Did he get upset about being in there when he was vomiting? Or is he just too busy to go to the bathroom? Sometimes 4 year olds don't want to be potty trained, and once they are potty trained, they get annoyed with having to do it all the time. The fact that he has tears means nothing.
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Original: If the neurologist has cleared him, that's great. So perhaps you should look more at the stress/tension angle. Is it possible that your son needs more reassurance after the hospitalization of the other child? Does "being sick" equal" getting attention" and make him feel more cuddled? Is his “under the weather” appearance possibly due to feeling worried? Could he benefit from some meditation (yes, they do it for little kids) and some other stress-reducing measures? There are great practitioners/teachers who work well with kids.
Have you completely ruled out an upper sinus infection? What about jaw clenching and temporo-mandibular joint pain (TMJ)? Both can cause headaches in the forehead. The first should have been eliminated by the pediatrician, but if it’s been “cooking” for a while, it might have been missed. And the jaw issues can be checked by a dentist, as well as the pediatrician.
Obviously, I'm just guessing here, and I'm not a doctor. My point is just that there are varied causes for this and a couple of simple things that can be done to see if they help.