Yes. There were some other disks in there as well, but I really don't want to risk screwing them up, in case it's the drive that's failing.
A failing drive is not very likely to screw up a floppy disk unless it does some sort of write command to the disk (which is not the case when an Amiga boots). The only reason it would "screw up" a disk is if the drive head was rubbing against the disk surface - which is not likely to happen. I would just try it - seriously you don't have much to lose and even if a disk was damaged you can easily find the .adf for that bundled software and re-write them to new disks when you get the system working eventually.
Is there no other way to check whether the floppies are broken or not? Weren't there two or three Windows tools that made it possible to read Amiga floppy disks without the need of a catweasel? EDIT: Found the tool I was thinking about. It's called adfread. I'll still need an Amiga floppy drive for it, of course.
I think you are confused. This tool requires two regular PC floppy drives, not Amiga drives. You must have two floppy drives installed in your PC.
You must also have a PC motherboard that supports two floppy drives in the BIOS - many newer motherboards do not support this "legacy" feature. Do you have an old 486, Pentium I or II hanging around to do this with?
Other than that the ONLY way to read an Amiga disk in a PC is with a CatWeasel or another hardware device like a Kryoflux that reads foreign disk formats on the PC.
The only other ways to boot into Workbench are with a hard drive or CD/DVD drive. There are PCMIA and SD card solutions for the A1200, but I'm not sure if you can boot from them. Another option would be to get a Subway or other USB interface for the A1200, but again I'm not sure if it's bootable.