Do a backup, using either Time Machine or a cloning program, to ensure files/data can be recovered. Two backups are better than one.Run Disk Utility/Repair Permissions (Applications/Utility Folder). Run until it reports no errors. If there is a recurring error, see the article on what permission errors can be ignored.
To get to the Recovery Mode hold down command - R during a restart.
Try setting up another admin user account to see if the same problem continues. If Back-to-My Mac is selected in System Preferences, the Guest account will not work. The intent is to see if it is specific to one account or a system wide problem. This account can be deleted later.
If the problem is still there, try booting into the Safe Mode. Disconnect all peripherals except those needed for the test. Shut down the computer and then power it back up. Immediately after hearing the startup chime, hold down the shift key and continue to hold it until the gray Apple icon and a progress bar appear. The boot up is significantly slower than normal. This will reset some caches, forces a directory check, and disables all startup and login items, among other things. When you reboot normally, the initial reboot may be slower than normal. If the system operates normally, there may be 3rd party applications which are causing a problem. Try deleting/disabling the third party applications after a restart by using the application uninstaller. For each disable/delete, you will need to restart if you don't do them all at once.
You are correct that this is not the work of malware. It's not that Macs don't get infected by malware, but that's rare and when it does happen, it doesn't usually lead to any noticeable symptoms. (See my Mac Malware Guide.)
As for what is actually causing the problem, try the suggestions in my Mac Performance Guide.
After backing up and repairing the disk, boot into Safe Mode and see how it runs. If you don't run into the same problems in safe mode, you have some third-party software that is not compatible with Mountain Lion.
If that is the case, we can help you figure out what it is.
If that is not the case, then I would suspect some problem with the hard drive.