@DerHeld,
Probably the best way to back up your old disks is using the ADF Blitzer utility.
ADF files are Amiga disk images used by WinUAE to read "virtual" Amiga disks. Using ADF Blitzer, you can read your disks and copy them to your HDD. You can also un-ADF them back onto standard floppies if you so wish, although be warned that you could be breaking copyright if you do so.
The advatage of ADF as a format is that you can connect your Amiga formatted HDD to a PC and read the disk via UAE. This will enable you to run your backed up Amiga games without having to use floppies.
Do a search for ADF on the Aminet for more info.