How to enable default file manager(nautilus) to handle desktop icons

As you already know - it is impossible, the functionality was removed from Nautilus.
By such actions GNOME developers do not respect user habits, so it is time for taking action by us, the users.

You have to install MATE DE with Caja (fork of Nautilus) by

sudo apt-get install ubuntu-mate-desktop^

then select MATE session in login screen and enjoy.

The below screenshot demonstrated all features of normal desktop integration between file-manager and desktop:

Here we have Mutiny panel layout and the following elements arranged by user:

• home folder icon;
• link to Dropbox folder (the folder itself is located in ~/Dropbox, controlled by caja-dropbox extension);
• desktop shortcut to Caja application copied from menu;
• pdf file with preview;
• simple text file;
• folder with Git repository inside (controlled by RabbitVCS extension);
• folder with Mercurial repository inside (controlled by TortoiseHG extension);
• simple folder with opened dropdown menu which demonstrates custom actions and menus from various extensions including custom caja scripts (usually located in ~/.config/caja/scripts) and actions configured using caja-actions-config-tool.

The screenshot of dropdown menu for empty desktop space is below:

Note: above screenshots may be overloaded by the stuff, but they demonstrate how flexible MATE desktop is nowadays. You can compare them with default GNOME Shell and GNOME FlashBack to make conclusions by yourself.