My daughter has many food allergies (most of which I figured out through experimenting with diet - hers and mine since I was breastfeeding). She also had trouble sleeping at night and later on had a fever that would not go away. The very day we quit eating dairy she slept through the night. It may be that, or it may be other things (for us soy, gluten, peanuts, corn and other things are also problems). It does sound like his immune system is weak, so probiotics and additional vitamins will be important.
The best way to check if food is a problem is just to start eliminating things and track how your son does. You can also look into having him tested for food sensitivities (IgG reactions). We had to find a naturopath (I can recommend on in the Denver area) to do the testing as that's not something doctors typically do. It showed us some foods that I wasn't figuring out were problems, but it also didn't show me everything that I had already figured out were problems, so there's still some experimenting to do on your part even with the testing.
You may also try giving him a large amount (for him) of vitamin C just to clear up the infection (you can't overdose on vit. C - it just flushes out of your system once your body can't absorb any more). My kids will drink Airborne (I like Walmart's Equate generic brand) and it gives them a ton of vitamin C in a way they can absorb and is tasty.
If you begin avoiding dairy, there are probiotics out there that are powdered, so you can just mix it in with juice or milk or even foods. We also have her on Nordic Naturals DHA Jr.
Good luck. It's a hard road, but just keep trusting your intuition and doing what's right even if when it gets hard!