It tells the type for the compiler, in case it couldn’t guess (in this case, it can guess it). Do note that case matters here: key and Key are not the same!
Generally, you don’t have to worry. The Arduino IDE prefers "", so if you plan to use it, go with the double-quotes. Otherwise it doesn’t matter.
In other circumstances, outside of the Arduino ecosystem, there is a distinction between "" and <>, namely that the latter is for system-wide includes, the former for local ones. The difference between the two is how the compiler will search for the files to be included. In our case however, it doesn’t matter much.
Nope, the order of Kaleidoscope.setup() and Kaleidoscope.use() should not matter. If it does, I’d consider that a bug.
That is harmless, indeed. If you want to get rid of it, you can remove (or comment out) the &TestMode line from Kaleidoscope.use(). Test mode is most useful for the factory, or when you want to test your keyboard. It is of little use in one’s daily life. Removing it will free up plenty of resources, and you can always put it back, should you need it.