Wallpaper displays incorrectly (partial screen), and will not let me change it to a different image.

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

IIf the above doesn't help try the folowing:

Where are your desktop photos located? They should be in the Macintosh HD - Data/Library/Desktop Pictures folder.

I renamed my computer to Toad Hall

Then put in a jpg file of the photo you want to use as your Desktop into that folder or a sub-folder with others you want to use. Then try to access them via the system settings as you normally do. I made sure the pixel dimensions of the photos are the same as my monitor's screen.

There has been a number of instances with Silicon Macs sleeping and having problems when waking up. The general workaround (not a fix) was to not sleep the Mac but use the Screensaver without sleeping the Mac and shutting it down at night.