I also solved the black menu problem. You can force JK to run in a window while in the menus. The game will still be in full screen and 3D acceleration with this setting, but will reduce to a window temporarily while in the menus. Still a bit annoying, but far less painful than having to alt-tab back and forth every time.
Just add the following command to the shortcut's target field:
Add -windowgui just after the quotes in the target field. Make sure you hit the apply button.