Wallpaper not switching with desktop
It's not obvious to me why the killall Dock command would help with this, but since it does it may be worth you doing a reset and restart of the Dock.

Reset and restart the Dock

If you go ahead with resetting and restarting the Dock you will need to re-add any non-default icons to your Dock. (I usually find it's a good opportunity for people to start afresh and only add things as they actually use them).

To reset and restart the Dock follow these steps:


Go to Finder
Press and hold the option key down
Now go to the Go menu
Select the Library option to open the User's Library folder. You can let go of the option key now.
Go to the Preferences folder (i.e. within the Library folder)
Find and move the com.apple.dock.plist file to the Trash
Now open Terminal (usually located in Applications > Utilities)
Enter this command killall -KILL Dock
Press enter


Remember you'll need to re-add some apps back to the Dock.

If this doesn't resolve your issue, it'd be worth also testing what happens when you're booted into Safe Mode and afterwards again. See below.

Boot into Safe Mode

Follow these steps to boot your Mac into Safe Mode:


Fully shut down your Mac
Restart your Mac
Immediately press the Shift key and keep it down 
Let go of the Shift key when you see the login window (NOTE: If you have FileVault enabled you may need to log in twice).
Take a note of what happens (i.e. switch desktops and take note of what happens with your Desktop Background)
Exit Safe Mode by restarting your Mac as normal
Switch desktops again and take note of what happens with your Desktop Background


Once you've booted into Safe Mode, let me know how you went.