A few more comments after reading over your post again:
I would expect him to be achieving better than average in a typical first grade classroom, at least in reading, but should also note that many elementary teachers don't assess beyond the grade-level standards, often because they're so busy trying to get the other kids up to speed (and some of them have mixed feelings about intense academics at this age), so he might actually be described as average only because they are not assessing beyond average. I would not expect him necessarily to be ahead in writing (as fine-motor speed is involved), or in math (as this is more closely associated with PRI and WMI, which are average on his testing). Though neither would it be surprising if he were advanced in these areas.
Clearly, his teachers are experiencing challenges with him, which is bound to affect his own view of himself as a learner. In addition to a comprehensive evaluation of him, I would hope that the information-gathering includes multiple classroom observations, not only of his behavior, but of the interaction between his environment and him.
To the question of whether lower processing speed can affect behavior: yes. Especially in someone who has relatively weaker visual-spatial skills, it may be overwhelming to try to keep up with all of the nonverbal communications that occur in complex, fast-paced, multi-peer interactions. On top of that, he is a verbal-strong kiddo, which may increase the likelihood of him reaching for a verbal zinger in an attempt to stay engaged with the group, which may not always go over well, especially in a group of six-year-olds. So you have a number of possible contributors to behavior: the symptoms relating to attention, impulsivity, high activity, social skills delays, etc. are all consistent with something in the ADHD category, but are also consistent with visual spatial weaknesses (sometimes called nonverbal learning disabilities). Both can also be reflected in delays in academics.
