You can buy ethernet networking test kits that verify cables are running and test entwork speed and soforth, like this one http://www.buytester.com/network_testers/6269.asp
As for simple hardware testing i've always found that having a good pool of working spare parts makes troubleshooting a lot easier. If say a floppy drive dies and you have a spare floppy controller ribbon and drive you can verify wether both these components are the problem by trying them leaving only the floppy controller on the board on the hardware side. Having the equivalent of a who pc in working spare parts makes finding a hardware problem easier.
I'm not really sure about software disgnostic, you could get sandrasoft to determine the details on a pc's spec and hardware. You could use something like norton system works to run windoctor and speedisk to try and opttomise running, maybe get a better defragmneter than the windows basic one and automate it. I'd get a few programs like that and leave them all to automaticly run weekly late at night or something if it's possible.