Yes. Low blood sugar can serve as a trigger for headache specially the migrane type headache.
Yes. High sugars in blood mean that the body is unable to use glucose. So, body and brain are starving for glucose. The body in response attempts to use glucose via other methods. Byproduct such as ketones build up. These can lead to a diabetic coma. Some people get headaches or nausea or sleepy or blurry vision or almost any symptom.
Probably not. Type 1 diabetes typically does not cause migraines or headaches.
No. Diabetes does not cause migraines but hypoglycemia-low glucose, can cause a headache!