It sounds like are are sensory issues going on, which can include postural issues and issues with bed wetting, and the advice PPs have given about getting a full neuro psych eval with particular emphasis on OT eval for suspected sensory processing disorder is spot on. Gifted testing will come on the side, as it were. The postural problems need not necessarily impact him in gym, depending of what they are doing.
Note that sensory issues, including the overexcitability in social situations you describe, are a very common "package" together with giftedness, but if it is impacting him in the classroom and in other social situations, they should be addressed.
