As for hospital trips--
---You can arrange through the resort for a doctor visit. My wife did this in our December trip for a sinus infection. Worked out well, charge was reasonable, and most of it was paid by insurance.
---You can catch a cab at any resort. I don't know where all of the area hospitals are, but there is a hospital at Celebration just a couple of miles outside WDW.
---There are first aid stations in all of the parks. Obviously, if anything serious, there are emergency vehicles and ambulances at hand.
---In any event, it wouldn't do much good to have a car back at the resort if you get sick/hurt while way across property somewhere.
As for the "crazy stuff":
---Some stuff you just can't worry about.
---You'll be in the same boat with many thousands of other guests. I assume that between WDW itself and the appropriate governmental authorities, someone has thought about this (they do have hurricanes in FL, after all) and there is some sort of
disaster plan. Evacuating huge populations over significant distances in a short time period is not very feasible in any event, so having your own car wouldn't make much difference.
As for staying at a monorail resort, while it is true that you only have monorail transportation to MK and Epcot among the parks, you also have monorail/boat transportation to a variety of activities and restaurants at the other resorts (including WL and FW).