How can I draw a background bitmap to a listbox? I'm using Series 60. Do I have to use some item drawer or derive my own list box instead of using the standard one?

I would say; you can't.

To be more specific, the listbox scrolling depends on copying areas of the screen from one place to another, which won't work very well if you have a background image behind the text. You would need a new listbox implementation built to support it.

Active Desk does it, so do I have to make an implementation of my own to mimic CAknDoubleGraphicStyleListBox etc?

Yes, you'd have to derive your own list box implementation. It shouldn't be too much work, though.