Startup on external screen

+do I have to run specific diagnostics? Or is that done automatically?+

As long as you hold down 'shift' it's done automatically.

+I ran Disk utility and tried to repair the MAC OS X disk. Unfortunately the macbook seems to restart randomly+

That's weird and shouldn't happen, I can't think what would cause that behaviour to be honest.

OK the Kernel Panic Log does help 🙂

It's caused by GeForce Kexts (kernel extensions) that can be removed and it should fix the Boot panic (and might even fix the screen problem) instructions on what do to here (Boot holding s to force single user/command line mode then enter the commands then reboot)
http://osx86.wikidot.com/known-issues

Take it slowly and type each command carefully and let us know how it goes.

If you use an external keyboard and mouse you can keep the laptop closed and only use an external monitor (do that most of the time here) it's called ClamShell Mode. Connect everything up then close the laptop lid to put it to sleep then touch a key on the external keyboard to wake it up.

So, you can't even start it up so therefore can't connect the externals and put it to sleep hmmm

"Kernel Panic at start up"
Have you Booted in Safe Mode to see what that finds and fixes ? (hold shift down at start and keep it held down for ages)
Can you Boot from the original Install DVD to run Disc Utility and Repair Disc ?
Have you tried moving/removing/swapping RAM sticks around to see if that impacts ?

Glad I could save you the cash and all's well now 🙂