What is the desktop manager in Linux Mint? I mean the program which handle icons on desktop, wallpapers, etc. -- I don't mean DE.
Desktop icons are handled by the file manager, which is
- Nemo in Cinnamon Edition
- Caja in MATE Edition
- Thunar in Xfce Edition
With KDE there's a special case, as there are no icons directly on the desktop. Here, desktop icons are held by certain widgets, which are AFAIK controlled by the desktop environment.
Wallpapers are being managed by the desktop environment itself, no matter which file manager.