[Emission]

Whether or not the tire retains its speed rating varies from manufacturer to manufacturer. For liability reasons (they aren't going to inspect every flat tire), many just throw and in the towel and tell you to buy a brand-new tire.

Use your judgement. As long as the belt isn't cut (most nails and screws don't cut the belt), the tire is still perfect.

My best top speed run was done on two patched tires without any worries.

(Kinda related subject... can aircraft tires be patched? Does the FAA allow it?)