I use this to convert video to the iPod, works perfectly.
http://www.videora.com/en-us/Converter/iPod/
I found that Quicktime Pro and others didn't convert AC-3 audio, which is why you might not have any audio on the video.
To backup your songs, you should just be able to burn a CD-R or DVD-R using the files stored on your computer, depending where the iPod syncs them to.