Yes. Type 2 diabetes is well know to run in families but type 1 diabetes does as well. Epidemiologic data shows that 1 in 400-500 people in the general population develops type 1 diabetes, but 1 in 20 people are at risk if a parent, sibling, or child has type 1 diabetes. However, research shows that genes don't tell the whole story and it suggests that unknown environmental factors also contribute.
Yes. We all have vulnerable areas of our physiology due to our genetic makeup. The key is to work with the factors we can control, such as diet, nutrition, and exercise. The entire field of epigenetics studies how we can compensate for these genetic tendencies in order to avoid the development of disease.
Yes. Type 2 diabetes, the more common adult onset type, has a 90% concordance within families, which means that 90% of a diabetic's close family members (kids, brothers, sisters) could expect to get diabetes eventually. Type 1 diabetes is not genetically predetermined.