How can I get a list of the running applications? I'm referring to the ones in the panel at the bottom of the screen.
The panel you are referring to is the GNOME panel. So this is a GNOME question, not a GTK question.
There is not a well-defined concept of "multi-window application" in GNOME that I know of. The panel task list is probably build by querying the window manager for the list of windows and grouping the windows by their "class" property.
There are also various window manager hints that must be taken into account, for example to ignore panels and other utility windows. In your place, I would look at the source code of the taskbar applet. There is maybe some documentation somewhere that covers the status-quo, but I do know where it would be.
How can I get a list of the running applications with GTK?