I don't think this is true... Having never upgraded an unused old iDevice,I'm pretty sure I have still been able to restore from backup without upgrading. But this is really old, so maybe they stopped that in a later firmware update?
It is absolutely true. If the OS itself is not corrupted, you can erase your data and restore YOUR DATA from backup, but that doesn't include the OS itself. If you do a "full restore" of the OS (and this isn't the first time I've observed the confusion caused by Apple's overloading of the term "restore"...in this context, "restore" basically means "complete reformat and OS reinstall"), you WILL be forcefully upgraded to the newest iOS.