http://reference.wolfram.com is the online version of Mathematica's documentation.

Not here. It redirects to the "Wolfram Language & System » Documentation Center" accessed by the Programming Could IDE Help.

The documentation has not been fully updated to reflect what parts of it have been implemented in the beta version of Mathematica Online/Programming cloud.

Source? No such declaration seen in the docs.

The documentation is for Mathematica

The documentation provided by the Programming Cloud "Help" menu is titled "WOLFRAM LANGUAGE & SYSTEM Documentation Center". I see nothing to suggest it is for Mathematica. If you have a source, please cite it.

Most of the things not included in the Programming Cloud

ControllerInformation is included in the Programming Cloud. The docs http://reference.wolfram.com/language/ref/ControllerInformation.html?q=ControllerInformation do not mark it "not supported in cloud", and Programming Cloud does execute it and return a result.

are things that aren't appropiate for a browser application. In this case, the joystick probably isn't very useful for a browser application where most of the computation is done server-side.

Gamepad is fully appropriate for a browser application. The docs say the gamepad is supposed to operate the in-browser sliders. The gamepad has no more to do with the server-side computation than does for example the mouse, which works fine.