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I have no idea what his school's policy is on testing, grade skips etc. Should I be starting with the school itself or the district board of ed? What questions should I be asking other than if they are willing to test my son?

I would start by calling your district and see what rules they have regarding skipping K, our district will not allow skips before K, they want time to observe the child in a class setting before moving forward.
After getting the details from the district I would approach the school next.
Warning, some schools get offended easy, so you have to tread lightly while making a firm stand for your child.

FYI. My MG DD7 loved K, being ahead of the class gave her the confidence boost she needed for the 1st year of school, DS3 on the other hand, already suffers from too much confidence. I doubt K will be as successful for him. My fingers are crossed.
I wish you luck and please keep us posted.