Is there a relationship with type 1 diabetes and obesity?

Nope. Absolutely not. It is a cruel autoimmune illness in which an overactive immune system creates antibodies which attack your (child's) pancreas until its dead.
More insulin. The only relationship is that if you are obese you may require higher Insulin doses to produce the same effect.
Depends on question. If the question is do chubby kids get type 1 dm more often than others, the answer is no. Given that about 1/3 of usa kids are heavier than they should be, the fluffy ones still get type 1 @ the same rate as others. Kids with poorly controlled dm can and do become overweight later do to a combination of issues. This is different from type 2 wher obesity is almost always a triggering factor.

Related Questions

Is it possible for an obese young adult to get type 1 diabetes and not type 2?

Sure. It could happen if you have the genetics for type one. It is possible also to get both, if you eat poorly. Eventually the poor lifestyle will lead to type two in addition to the type one.

Is it possible for an obese young adult to develop type 1 diabetes and not type 2?

Yes. Obesity is known to increase risk for developing type 1 dm in both adults and kids. Some adults with type 2 diabetes also have antibodies for type 1 diabetes. Some would argue that both types of diabetes are the same disease on either side of the spectrum. We do sometimes diagnose children with "double diabetes" when they clearly have antibodies but also significant chronic Insulin resistance.