Are there affordable fan-cooled PSU's whose fan does shut down when the computer is put in Sleep/ShutDown mode? Fanless PSU's are pretty expensive
If you look at SPCR's reviews, you'll see that there are fairly affordable PSUs whose fan only starts when they work hard.
I leave my PC on 24/7 because I fear that shutting down/restarting hard drives increases the risk of mechanical failure: Is this fear confirmed by experience?
Not really unless perhaps you shut them down twenty times a day or something. Drives fail in any case and most failures are unpredictable. Do your backups properly and you'll be fine.