MacOS – Macbook and external hard drive : A better solution

You could install a retail version of Windows on the Mac and migrate your data (which you would previously have backed up) onto it. Just mounting a Windows install that had been made on other hardware is pretty much doomed to failure. The correct hardware drivers won't be present and the existing ones wouldn't be correct for the hardware, for starters.

disk3s3 is the extra partition on the time machine disk which OS X is complaining about. From what I have read (https://discussions.apple.com/message/19675983#19675983) this is a partition called "Boot OS X" which might have been installed on your time machine disk around the time of upgrading to 10.8.2. According to the linked article, this partition contains a copy of the Recovery Partition (http://support.apple.com/kb/ht4718) which you can boot from in the event that you cannot boot from your primary hard disk. This will only give you access to Recovery Mode, not a full OS installation.

I have observed that this partition is not mounted when you plug in the time machine disk (use the "mount" or "diskutil activity" commands to see this) but instead appears to be mounted when Time Machine finishes a backup. I don't know why it does this.

When you "eject" the disk from the Finder, you probably need to wait a few seconds before unplugging it. When the drive icon disappears, this probably means that the main time machine volume has been unmounted, but maybe it is still unmounting the recovery partition. This is likely the cause of your error message about having unplugged a disk without ejecting it. If you are paranoid, type "mount" to see all the mounted volumes, and verify that the time machine volume is no longer mounted before unplugging the disk.