Desktop icon positions not remembered between sessions

It's possible that a file recording the position of icons on your desktop cannot be modified due to wrong permissions.

To identify such files, run find . '!' -user $USER.

In your case, the output included ~/.local/share/gvfs-metadata/home. Changing the ownership of this file to you should fix your problem.

In general, very few, if any, files and folders in your home folder should be owned by root.