Desktop is missing

Since you can access System Preferences, can you go to Users & Groups and create a new user? See if the new user has the same issues at login. If not, move your files from /Users/oldusername to /Users/newusername and then delete the old username.

Do you have a backup you can restore from?

I'm assuming the new user is an administrator.

Logged in as the new user, launch up Utilities > Terminal.app, and then enter a version of these terminal commands (you'll have to substitute in actual usernames for olduser and newuser):

sudo mv /Users/olduser/Desktop/* /Users/newuser/Desktop/

sudo mv /Users/olduser/Documents/* /Users/newuser/Documents/

sudo mv /Users/olduser/Downloads/* /Users/newuser/Downloads/

sudo mv /Users/olduser/Movies/* /Users/newuser/Movies/

sudo mv /Users/olduser/Music/* /Users/newuser/Music/

sudo mv /Users/olduser/Pictures/* /Users/newuser/Pictures/

The reason we're moving individual folders instead of just the entire user folder is there is clearly something corrupt in your old user's configuration, so what to move only the files and not the settings... otherwise, the new user will be messed up as well.

Then, change ownership of those files to the new user

sudo chown -R newuser /Users/newuser