MacBook – Wallpaper settings keep resetting

Having a bit of trouble following your description of the situation. It'd be helpful if you used actual path names instead of "The one in my user folder" type descriptions.

You have a backup right? Ok then...

The sidebar link is just a shortcut, so it doesn't really mean anything by itself. Fix that last. Just delete it and then re-add it.

Run a disk check and permission repair if you haven't already.

If ~/Desktop doesn't exist and you're sure it was some how renamed .tsclient, use terminal to move/rename it back. I think its more likely that your ftp client uses that as a temp location and that the files there are only partially downloaded though.

If ~/Desktop does exist then just move the folder contents back where they belong not the folder itself. Then delete and re-add to the sidebar

I think they're working as expected. Center puts the image at the middle of your screen, without changing the aspect ratio (an image larger than your screen will be downscaled, but a smaller one won't be upscaled). Because your screen is a 4:3 ratio, and the wallpaper is 16:10, you get blue bars filling the rest of the space. Similarly, Fit to Screen scales the image proportionately so that width matches that of your screen, but it does not change the aspect ratio, which is why you still have blue bars.

If you want the image to take up all of your screen without any blue bars, try the Fill Screen or Stretch to Fill Screen options. Fill Screen scales the image proportionately so that it fills the entire screen, and Stretch to Fill Screen scales the image to fill the screen, but changes the aspect ratio so the entire image is on-screen (i.e. you don't lose any of the image boundaries).