It is virtually impossible to help you to come to an accurate diagnosis online. It might help to recognize that most people show a combination of various patterns, i.e. you could have concurrent kidney jing deficiency, kidney yang deficiency, spleen qi/yang deficiency, and damp/phlegm accumulation. Any developmental issues are related to the kidney jing, so for example not all the baby teeth coming in certainly suggests at least some trouble there. Cold is usually a yang deficiency sign, but not always and it could relate to kidney, spleen, or heart yang for example. Mucus could come from exterior cold, a spleen or lung yang deficiency, poor diet, or some combination of all those factors.
If he had 10 visits of acupuncture with no improvement then whomever you saw likely isn’t able to either properly identify the patterns your son is presenting with or doesn’t know how to treat them.
My best suggestion is to find another acupuncturist in your area. They should be better able to both diagnose and explain your son’s condition.