Create a new user. Press ctrl-alt-f2. Log in as that user.
$ touch .xsession
$ ln -s .xsession .xinitrc
$ chmod u+x .xsession
Edit the .xsession file and make it contain this
Now get a graphical login screen up and log in as that user. You will find that Firefox launches automatically and nothing else is running. However Firefox won't fill all the available space on the screen and there is no way to make it do so because there is no window manager running. Firefox has no command line option that allows it to be launched in full screen mode. So although you've got an environment with just the one application running, it's not very nice. If you close Firefox the user is logged out.
If you edit the user's .xsession file so it looks like
then log back in as that user you will find Firefox is still the only application running and it's still not using all the space on the screen. However you now have window decorations because the metacity window manager is running, so you can maximise the window in the normal manner. If you minimize the window it vanishes, but you can press alt-tab to get it back.
So that, along with having GDM log in automatically as that user, is basically how you can give a user an environment with just one application. If the application you want to run can be somehow made to launch in full screen mode, either by passing a command line argument or by putting something in a configuration file that it looks at in the user's home directory, then that makes it all a bit neater.
If there are problems with the video output before
the operating system has started to load then there is a problem with the hardware. (Posting in the Hardware section isn't going to get you help with that because it's not a case of getting the hardware with Linux, it's a case of the hardware being faulty.) You should be able to determine whether the fault is with the monitor or the video card by attaching an external monitor.
I don't see why the laptop couldn't be used as a headless router.