Are there tests to confirm the doctor's diagnosis of child with diabetes?

Yes. Generally, if this is type 1 diabetes, a test called the hba1c can confirm the diagnosis as it determines if there's been a chronic elevation in blood sugar. In children, the presence of sugar and ketones in the urine are also tell-tale.
Yes, there are but.. ...Usually not necessary in most cases of type 1 diabetes. Most of affected children are diagnosed in diabetic ketoacidosis or are showing characteristic symptoms such as weight loss or frequent drinking/urination. Type 2 diabetes in children is usually associated with obesity and Insulin resistance. Elevated fasting blood sugar and hemoglobin A1c levels confirm the suspicion.
Yes. There are blood tests that can measure the body's handling of blood suger and the relationship to Insulin and other metabolism that show the symptoms. If done, these tests are 100%accurate. Check with your doctor.