Hello! I don't know your name so I'll just call you AD for Agony's Daughter.

-First, let's remove everything associated with Apple. Uninstall iTunes AND QuickTime. Make sure there is nothing left of those programs.
-Do a reboot after uninstalling both of them.
-Once you've rebooted, try downloading the latest version of iTunes and then install. It should install QuickTime also.
As far as your music on your iPod, you cannot actually edit what is on your iPod.
Once you get iTunes reinstalled, you will be able to edit all of your music and playlists and sync it to your iPod.
The problem with syncing 1 iPod to 2 different computers that have iTunes, is that the synchronization is only 1 way. Meaning, only stuff in your iTunes gets synced to your iPod, not iPod to iTunes. So if you plugin your iPod to your mum's computer and she doesn't have all the same music and playlists as you do on your laptop, your iPod will have all it's music and playlists erased until you plug it back into your laptop and resync.
My suggestion would be to do the first 3 steps up top, and once you have iTunes working again, before you even plugin your iPod, build your iTunes library and once you're sure everything is there, go into iTunes options and check the option to "Manually Sync Music and Playlists". This will allow you to plugin your iPod and add/remove music, add/remove playlists however you want without the iPod automatically syncing and doing what it wants on it's own. Once you get everything organized how you want it in iTunes, then you can UNCHECK that option and let your iPod automatically sync.
Alternatively, you can use a 3rd party program to synchronize your iPod.
There are 2 programs in particular that I like.
MediaFour X-Play:
http://www.mediafour.com/products/xplay/MediaMonkey:
http://www.mediamonkey.com/Good luck and let us know how everything is working out!