Is it possible to run graphical applications such as Firefox without installing a desktop environment?
Well, Firefox REQUIRES X server. What you would need is to install Firefox with the help of the package management - it should install the bare minimum of the packages, including the X server. Then, you just need to run the X server without the desktop environnment. Basically something like:
$ X
Then you just start Firefox in this X server:
$ DISPLAY=:0 firefox
You can switch from the X server and the framebuffer by using CTRL+ALT+F1 and CTRL+ALT+F7.
Hope it helps.