Internet research suggests that early October is still a hurricane risk for Florida. There is a small chance that a hurricane could hit while I'm there. Disney property seems pretty safe and well built although I'd trapped inside the hotel room during the hurricane. It would be a bit too much for me but I think I'd still be okay (not break down totally panicking) if I had food inside the hotel room during the hurricane. I also read there are back up generators so if power goes out, I'd still get power in the hotel room. Since you're adverse to any non-sunny and any chance of any eventful weather, maybe later October would be better for you.
Here are the weather stats for Orlando- rainfall drops dramatically from September to October so I assume most of the rain will be in early October-
http://en.wikipedia.org/wiki/Orlando,_Florida#Climate
I now wish I had planned the trip for the last two weeks of September to increase my odds of experiencing thunderstorms but my friend can't get any time off in September so October it is.