Is the harddrive installed?
First off, since you CAN boot via f loppy, you KNOW your laptop works.
Since you DO NOT see a harddrive on the desktop when booting with a floppy and the disk tools give you an error. One of 3 things must be happening.
1. NO HARDDRIVE is installed
2. Hard drive is corrupted and needs to be repaired.
3. Hard drive has failed and is useless.
Before you voice frustration about this Mac, keep in mind, two things. Its 11 years old. As you said, it was cheap. 11 in computer years is ancient. Think about it, even a car that is 11 is considered old. That said, you can get this to work.
Verify you have a harddrive inside. Its possible the previous owner removed it. If you have one, then you need to get the model and make and then determine if your boot floppy is the proper (meaning best for this task) version to handle this model and make of Mac. Since its so old, the biggest problem will be finding this older software.
Early macs were plagued by random errors from bad ram,. disk errors, etc. Your mac is old. Random crashes with a floppy based start up WILL happen. The drive itself could use cleaning, etc.
I suggest you verify the drive, and work from that point.