backup problem

Are you sure you are authorizing your computer for the correct ID? Go to your music library for the problem song(s) such as the one listed in the error message, right-click on the song and select Get Info, on the Summary tab look for Account Name. This will be the Apple ID that needs to be authorized on your computer. (You can view the list of items the warning is referring to by clicking on the "+" sign on the lower left of the warning message.)

Sorry, my mistake, it's the songs on your phone that you need to have the ID for, not the ones in your iTunes library (they aren't in your library yet). There must be a number of them on your phone that were purchased under an Apple ID that your computer is not yet authorized for. You may have purchased them earlier under an older Apple ID or they were purchsed under somone else's ID. In either case, you won't be able to transfer them to your library until your computer is authorized for the Apple ID used to purchase them.

As for the backup, it doen't include your iTunes media (like all your music, videos, etc.), so it would be considerably smaller than the 14GB being used on your phone. The backup is only there to restore your phone. It doesn't back up the iTunes media because that is already on your computer and can be restored to your phone from there. It also doesn't include your email and certain other data. To see what is included in the backup, see http://support.apple.com/kb/HT4946.