Could I still run graphical based apps like Firefox & OpenOffice if I didn't have a GUI (GNOME, KDE, etc.)? Could I just use X11 and extensions, but still keep a CLI?

I'm pretty sure the answer is no.

Im pretty sure the answear is yes . Tho as to why you would want to do that, i dont know. There is little memory savings if thats what you think of.
XFCE uses very little memory imho, and there's still many ways to save resources when using any gui. I'd look up a gentoo forum and ask there.
I remember seeing some screenies where there was only like IRC program opened and a bash/shell/cli if u will.

How does one overlay graphics, such as a Firefox window, over a full-screen text display?

You'd need something that actually handles the x-windows. If you don't believe me, start up your Linux box, and hit Crtl+Alt+F1 to drop into a command shell. Login and then run xclock. You'll probably get an error about the Display not being set properly.

Like TranceEnergy said, XFCE is fairly lightweight, as well as E17. I personally find GNOME to be pretty resource friendly, provided that you don't add any large panels or themes.