Are you sure that the non-caching behavior is not associated with one or two particular websites? The headers sent in the HTTP response of a webpage can force a page to be non-cached.
Edit: Sorry I noticed this after a re-read: "Checking the profile folder the "Cache" folder is missing as well all of a sudden." Yeah, that's not a good sign.
Re-edit: what extensions do you have installed for Firefox?
Have you checked to see if the problem affects different users on your machine?
You could try specifying explicitly a cache directory using the
Browser.cache.disk.parent_directory value in about:config
http://kb.mozillazine.org/Browser.cache.disk.parent_directory)
If that fails and you are desperate, a reinstallation of FF would most likely fix the problem.