... will release some from "storage" aka fat.
Actually the first storage glucagon releases is liver glycogen, not fat. When you eat and take insulin, your body restores blood glucose first, then switches to liver and muscle glycogen, then switches to storing as fat. Glucagon inhibits the action of insulin and prompts the liver to convert its glycogen stores to glucose, which is then released into the blood stream where it can be taken up by the brain and (assuming the presence of insulin or exercise) muscles. Glucagon will break down fat and release free fatty acids (and ketones) only once the liver glycogen stores are depleted.
Like you said, the beta cells dysfunction makes the alpha cells somewhat dysfunctional - though they are not technically broken; the system is just broken by the beta cells. The same is true of delta cells, and PP cells, and also some other endocrine cells, like the adrenal glands. Delta cells produce GHIH (growth hormone inhibiting hormone) which (surprise) inhibits the action of growth hormone. The most obvious effect of growth hormone is the increased insulin resistance in the morning known as dawn phenomenon - again, happens in people without diabetes as well, but it doesn't cause problems because the larger system isn't being strained by dysfunctional beta cells. Epinephrine (or adrenaline) is a similar story. Cortisol also has effects on blood sugar. Endocrine function is enormously convoluted, and everything is secondarily or tertiarily affected by the dysfunction of beta cells.
However, as far as I know the majority of people with T1 have absolute deficiencies of insulin and amylin and nothing else. However, there are still not that many studies done on T1 folks, and science is constantly evolving, so we may find out next week that type 1 also causes a definciency of something else. And this ignores the increased likelihood for other autoimmune endocrine disorders i.e. hashimotos.
Between the time scientists isolated amylin and scientists were able to create injectable amylin, most health care providers who worked with T1 people thought that, when amylin became available, everyone would be on it. Then it came out, and they found that it had almost no effect on control over a large population and caused a lot of stress because it meant three more injections every day, caused people to feel nauseated, relied on people to adequately assess their carb intake before eating in a new way, etc. That isn't to say that it's not good for some people - it can really help some people with constant hunger (and i think other things, also, but i don't know much more about it.).
I am kinda a layman's endocrinology geek. I did this for a class last semester. I think it's fascinating.