A GRUB error would not cause problems with the file manager. If the system is booting, GRUB has done its job. It is not uncommon for installers to report minor issues with installing GRUB when it is actually installed and working, which appears to be the case here.
Can you post the error messages or specific problems you have with the file manager and software manager? "do not work correctly" could mean anything
If you need to reinstall, just run the installer from the DVD again and when it asks where you want to install, replacing your existing Mint system should be one of the options.
"Insanity: doing the same thing over and over again and expecting different results." (Albert Einstein)