This is obviously a very frustrating problem for your family. While it is almost impossible to give you an answer from an email, we can at least give some explanations of the cause of cough and why these may or may not be the source of the problem you are facing.
Coughing usually comes from an upper or lower airway source. Let's eliminate the lower airway sources first. Asthma is the most common source of chronic cough. It is most common at night or the early morning hours and is sometimes worse with activity. Asthma can be diagnosed in an 11 year old fairly easily and this is not the case here. Other lower airway causes are more problematic, but also present in a very different way. These can include pneumonia, bronchitis and a foreign body. All of these are fairly significant medical problems, relatively easy to diagnose and are almost all worse during the night.
Upper airway causes are much more difficult to diagnose. Probably post nasal drainage from allergic conditions, nonallergic inflammation and sinus inflammation (with or without an infection) are the most common culprits. These conditions, most of the time get worse during the night because the drainage is worse when lying down and will pool in the back of the throat and wake up the patient. This is not absolute like pneumonia, but does occur in the majority of cases. Sinus x-rays help when the inflammation reaches a level that can be seen. Sinus CT scans are better to look at the lining of the sinuses and give a fairly good idea if the sinuses are involved. Skin testing does not tell you a cough is due to allergy, but does tell you if the child is allergic or not. Negative skin testing eliminates allergy, but not a nonallergic inflammation seen as a result of a previous problem, such as a sinus infection.
A common cause of a cough which gets worse when the child is awake and goes away when they are sleeping is psychological or a habit cough. This diagnosis is only made when the others are eliminated. It is a common problem in children who are upset for one reason or another. It is not that the child coughs on purpose, but that the cough that started for a physical reason becomes a habit and continues even when the physical reason for the cough no longer exists. Tests show nothing, the cough goes away once the child goes asleep and if the focus on the cough gets less and less, it goes away. What does not go away is the reason for this behavior which is not such an easy problem to deal with.
I hope these explanations of chronic cough help you to look at some options for investigation.
I overcame the chest pains from coughing by learning to suck the mucus down and get rid of it before it went down and had to be coughed out. Doesn't help when you're asleep though. I had allergy tests that only showed pollen becuase it didn't contain many food tests. i'd see if antihitamine helps - it would prove an allergy if it does. I'd also use a liquid form and start with a small dose - the mucus can sry out and be even harder to get out with a larger does. Also take lots of water to keep the mucus liquidy. good luck,