For some stupid reason I excecuted sudo chmod -R 755 /. Now none of my software will run. Can I fix this without reinstalling? And if yes, how?
there's no real undo to chmod. If yours is a reasonably new install we could give you a list of all main files/directories and you could try to restore permissions one by one, but sincerely that would be a hell of a job.
You could better boot from live CD, backup all your stuff and re-install.