Mouselook mode doesn’t work well in a browser on OSX, that’s one of the issues still outstanding. (It does work fine in standalone, as you observed.)
The OSX runtime app doesn’t quit itself when you close the last window. This follows the general OSX convention: most apps don’t quit themselves when you close the last window. Although now that you mention it, I’m not sure that command-Q does the right behavior when you apply it to a window; it might only close the window instead of quitting the app.
David