iTunes – How to Recover Deleted Backup

With iOS devices there are a lot of issues that can be happening between iTunes and the iPod. Backups can take a very long time depending on the amount of application data or the size of the camera roll at the time of the backup. Troubleshooting these issues go from very simple troubleshooting steps such as updating iTunes and restarting your computer all the way to the more advanced troubleshooting steps. If you are getting an error code, this can help you determine the exact issue, there is a really great article for troubleshooting the iPod on apples support site @ http://support.apple.com/kb/TS1275

Some steps that you can take to help isolate the issue would be trying a different USB cable (if you have one) or different USB port (should be USB 2.0), if you are connecting to USB through USB hub or extender try plugging it directly in to the computer, creating a test windows user account can also save a lot of headaches by letting you know it is something specific to that user account, and trying to sync music on another computer. I would try walking through that article and if your still having issues after that to try contacting apple or setting up an appointment at an apple store to make sure there is nothing wrong with the actual hardware of the iPod. Best of luck

You may be stuck on iOS 7 if you don't have a way to get that software from Apple's servers. Typically only registered developers can download several versions of software, but if you search - there are many places that archive these software ipsw files. It's always a bit of a risk downloading software from a place other than the vendor, but I don't know of a better option.

Typically, when a new OS is released, iTunes will only download the latest software if you don't already have an older version of the software stored locally. Is there any chance that computer (or another computer) downloaded iOS 6 and still has it on the hard drive? If so, you can decline the offer to upgrade and restore the device from that version of firmware.

As was covered in the comments - there is no restriction to load iOS 6 backup on iOS 7. Also - this is more about the computer and iTunes having the old iOS file already downloaded than something that's inherent on the device. Any change you have a backup copy of the iOS files iTunes downloaded?