These slots are only ever attached to event handling mechanisms within
the class itself, they're never used externally. Because of this, we can
make the functions private.

This also removes redundant usages of the private access specifier.

Resolves trivial compiler warnings.

Instead, we make a proper registry class and house it within the main
window, then pass it to whatever needs access to the loaded hotkeys.

This way, we avoid a global variable, and don't need to initialize a
std::map instance before the program can do anything.

This makes clang-format useful on those.

Also add a bunch of forgotten transitive includes, which otherwise
prevented compilation.