At first make sure cosmic is really installed. Select it at the Login Screen and login. Then take a look at the config file for your desktop manager. Is it sddm, try:
sudo nano /etc/sddm.conf.d/autologin.conf
Then change Session to cosmic
[Autologin]
User=john
Session=cosmic
DE will be deleted with pacman, try this for kde:
sudo pacman -Rns plasma kde-applications
To install a DE use pacman, try this for xfce:
sudo pacman -S xfce4 xfce4-goodies
I hope this will help.