Q: My Mac does not boot up, how to solve it?i

The image is a kernel panic

Mac OS X: How to log a kernel panic - Apple Support

The fact that it happens in recovery mode & when using external disks points towards the hardware (provided the external OS was known to work). If the external is a 'clone' of the internal try restating with a new OS X install on the external - a third party extension could be part of the cause since a safe boot works. A installation from another Mac may not work correctly, unless the hardware is very similar, use caution when testing like that, each install can have custom extensions & drivers that are hardware specific.

I would check the RAM since it's one of the few things you can remove & test.

MacBook Pro: How to remove or install memory - Apple Support

Or look around for guides or videos…

http://macs.about.com/od/usingyourmac/qt/macbook-upgrade-guide.htm

If you have multiple DIMM's installed remove & test with just one fitted (follow Apple's instructions for the order of installation), otherwise just check the RAM is seated correctly.

If you can access and remove the HD try booting with the HD removed, just use the external.

When woking inside the Mac take antistatic precautions…

http://www.computer-memory-upgrade-stick.com/installing-ram.htm