1). Yes, you can set up Orange broadband on your laptop. Your mum should have a CD from Orange with all the necessary settings. Use that. You will need to know your mum's login and password details. Note, however, that the PC and the laptop will not be able to connect at the same time using the same 'phone line as there will be insufficient bandwidth.
2). Again, you can use your mate's Tiscali CD to put Tiscali settings on your laptop. You will have to use your mate's login and password details. Again, you will not be able to connect at the same time as your mate on the same phone line.
3). If you are using Windows XP you can use the built-in backup wizard. Go to My Computer and right click on Local Disk (C:). Click on Properties and then choose the Tools tab. You will then see an option to Backup. Ensure your external HDD is connected, click on Backup Now and work though the wizard. Once everything is saved to the external drive I do NOT suggest that you delete everything on the laptop. Leave it where it is. The whole point of a backup is to have an extra copy of your data in case your laptop crashes. But if you delete from your laptop and the external drives dies, you will have lost everything!