Welcome to health care magic
I can understand your situation. I appreciate your effort to seek online support.
As per your description, your daughter might have some allergic phenomenon which is causing cough
and itching sensations in hands and foot. Increased severity of cough on lying down could also occur due to pulmonary (lung) pathology. Consult a general physician for detailed work up and investigations like Complete blood counts, X ray
Don't worry and relax.
Hope it helps
Dr. Gourav Monga