Not sure about LA but we are in the height of flu season and my kids have both had runny noses and coughs.
As for the excema and allergies, do you suspect a hidden allergy? Have you considered what he is eating, could it be wheat? Could it be soy (which is hidden in most processed foods)? Could it be hidden milk? (so hard to find it all)
It takes a while for the proteins to get out of their system. For us, usually about a week, that would heal the diaper rash and digestive ills.
My other suggestion is what are you using for bath wash? Johnsons makes my dd break out, we use California Baby brand with huge success.