Hi gell. These may both be the same problem, or may be different. The floppy drive could be faulty, and fortunately are very easy to replace and can cost as little as £5 click here. It is also possible that the cable is faulty or not properly connected.
The fact that having a floppy disk in the drive doesn't interfere with booting could be because the BIOS is set up to boot from the hard drive before the floppy.