S.C.
I would guess that the diarrhea, cough, runny nose etc are all symptoms of some type of virus he's caught rather than allergies. I would bet that you hit the nail on the head when you said that the diaper rash is from the loose stools. And that the loose stools are from not eating and only drinking juice.
There probably isn't a whole lot you're going to be able to do for the cold (cough, watery eyes, snotty nose, etc). There aren't any type of cold medicines that little ones can take. Just make sure to use a saline spray in his nose. If he doesn't know how to blow his nose, you might have to use the bulb syringe (snot sucker) to help clear his nose. Sitting in the bathroom with the door closed and the shower on as hot as it will go should help too. You can also use Vick's on his chest.
For the diaper rash, get a good diaper rash cream. I like Balmex. Avoid wipes since they have alcohol in them and can burn. Use a warm wash cloth or a piece of old t-shirt to wipe his bottom. Layer on the cream really really thick and change him as often as you can.
For the lack of appetite..wait it out. You can't really do much else. Offer him bland foods (BRAT diet) and that should help a little with the diarrhea. Also, no more juice! It's just going to make things worse. Try some Pedialyte or watered down Gatorade or Powerade.
I'm dealing with a couple sick kids of my own right now :( Let's hope they all feel better tomorrow!