Yes. An eye doctor can tell if you have diabetes because he can see the changes in your eyes that is caused by diabetes. If your eye doctor say you have got diabetes, then you need to see an ophthalmologist and let him test your eyes.
Yes they can. Because diabetes can cause diabetic retinopathy and there will be some changes in your eyes if you have diabetes. If he told you have diabetes, you need to be checked again by a physician.