How to restore a 10.1 backup to an iPhone with 10.0.2 without updating to the newest iOS available on itunes (currently 10.2)? Specifically, my backup is for 10.1 but the existing iOS on the phone is 10.0.2. When I plug into iTunes, it does not even show my backup from the "Restore Backup" option.

While I would prefer to have the full backup, the firmware is the most important to me. Does anyone know if there is a way to do this?
Backup and restore doesn't change firmware, only user data. That is how iTunes is designed. 

Same with the regression, backups do not go backwards. Only forwards. You can restore a backup from an older version to a newer version of iOS. 

Best case, you export the data to a computer or online service and find a way to jailbreak or buy a device with the older OS. Then you could import the data as opposed to the backup. 

Also see:


Can I install any iOS version I wish by picking and choosing which firmware to restore?