You didn't go into BIOS to run the diagnostics. You ran them from a hidden partition on the hard drive. 
You'd get that error about the diskette drive if you didn't have a floppy disk in the drive. So you might want to run that diskette drive test again with a formatted floppy in the drive.
So the question is still about noise. I'd back up on external media asap in case the hard drive is making the noise. Power off, unplug and press/hold power button for ~15 sec. Open the case and very gently check that the CPU fan and video card fan (if it has a video card fan) are free of dust and their blades turn freely. Use canned air to blow out all the dust bunnies from all fans, CPU heat sink (caution: heat sink may be HOT!) and case air vents. Make sure the 4 screws holding the heat sink in place are tight, but do NOT loosen them. Also check that wires or cables not are hitting any fan blades.
If you're careful you might boot up with the case open to see if you can find out where the noise is coming from. It might be the heat sink fan, video card fan or power supply fan or a drive.
After you fix the noise issue, then you can work on the wake issue, which might be software and unrelated to the noise. Aside from not waking (and the noise) does it cold boot normally and work ok?