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I am Dr. SS. I will try my best to help you.
Since when he is pale looking?
Does he have fever?
How is his food intake. Does he take meat, fish, vegetables?
Coughing for 8 weeks should be checked by doctor.
There can be an infection.
Allergy can also cause cough.
In infection the child can lose some appetite. Please try to give him adequate fluid.
Pallor can come if he is anemic or have low hemoglobin in the blood.
Anemia can cause him to be pale.
Infection can lead to anemia.
Sleep problem can occur from coughing.
So he can feel drowsy at day time.
I would suggest to get him examined by doctor.
He is likely to get a blood test.
Doctor would check hemoglobin, blood counts.
Eczema is not directly related.
Some immune mechanism can cause allergy, eczema.
If you have any query, please feel free to ask.
You are welcome.
The cause is not often clear for eczema.
There can be genetic or environmental factors.
You need to check whether any thing aggravates his symptom.
He needs to avoid that.
Allergy, eczema can be associated.
Supplements would not help.
Often allergy gets less when the child grows up.
You need to avoid those detected by prick test.
Do you have any other query, please ask.