PC shut down or sleep over night

It depends on the type of sleep mode set.
The first sleep mode, will shut down the system fully, except the ram and it will store all the current running in there until you wake it up from sleep mode. The drawback with this mode is obvious if you have a power cut during the night, then all the data is lost.

The other mode saves and image to the hard drive of the state the system, os was and any other programs running at the time, and then sends the rest of the computer into sleep mode clearing the memory. Then on power up reloading the saved running os and programs that were running from the image it made to the hard disk drive.

I always do sleep - minimal power draw- and have it on a small UPS for those power outages.

Going as a corollary to this question, is there any harm to shutting down / powering on a computer multiple times a day? I didn't think there should be, but things like this make me wonder.

There is a train of thought that you should always leave them on to reduce the heat/cool swings. There is a small amount of expansion and contraction with each cycle that can create microfractures in the traces or at solder points.

I did have an 8800 gtx start failing and the temporary fix that worked was to back it at 385F for 8 minutes, the thought being to soften the solder enough to make better contact again. I limped it along an extra 18 months periodically doing that.