I think keeping the right attitude is important. For the hospital stay, for example, find a place farther away to park, then bike or walk to the hospital. The exercise and time outside will help you feel better and give you time to clear your head. And bring a lunch since eating good food will help you feel better as well. Make time to do something that's calming/relaxing and don't let the stress affect you constantly.
After something like a hurricane, assuming that your house is intact and everyone is OK, you just have to think of it as an additional challenge/ being forced to be more mustachian for a while. It shouldn't take any electricity to get to work, even if you drive. If roads are blocked by trees, that's a perfect time to bike! Biking to the grocery store is also great when you don't have power because you can't carry as much as you normally would in a car, but you can't store stuff at home anyway since the fridge won't work. Of course that assumes the store has power and food, which are not always the case. You can probably get creative with ways to prepare food, if needed. If you have an electric stove you can instead cook on your gas grill. You can even boil water if you take the grates off and put pots directly on the metal shields over the burners (depending on the design of the grill). You could bathe in said hot water if you needed to. If you don't have a gas grill or stove, you could build a fire in your fireplace or in your backyard and cook stuff in tinfoil and sit around roasting marshmallows. There is almost always something you can do to replace a modern convenience with something less convenient but more interesting and fun!
Instead of thinking of these emergencies as bad events that are stressful, think of them as challenges that demand problem solving and flexibility. Make the most of what you are given. All the things you mentioned are unavoidable, and the only thing you can control is your response, so don't let yourself be consumed by stress.
