This is caused here:

		mw.loader.using( 'jquery.ui.draggable' ).then( function () {
			gui.draggable({
				handle: header,
				containment: 'window',
				start: function ( event ) {
					if ( event.toElement.id !== 'proveit-header' ) {
						dragged = true;
					}
					gui.css({
						'right': 'auto',
						'bottom': 'auto'
					});
				}
			});
		});

event.toElement is undefined in Firefox. The intention of the code is to prevent clicks on any of the buttons in the #proveit-header right after a drag has occurred. event.toElement.id can be replaced with event.originalEvent.target.id, and it works for me in Chromium 57 and Firefox 54. event.originalEvent is the original jQuery mousedown event; .target then gets the element which was actually clicked.

@Uglemat Awesome work!! I merged your changes on https://gerrit.wikimedia.org/r/#/c/347717/