I think the problem with your touchpad is it's detected as a mouse for
whatever reason.
To hide the icons, you need to e.g.
gsettings set org.gnome.nautilus.desktop computer-icon-visible false
the other prefs are
org.gnome.nautilus.desktop home-icon-visible
org.gnome.nautilus.desktop network-icon-visible
org.gnome.nautilus.desktop trash-icon-visible
Not sure if the gsettings package is installed by default.
...or goto settings > advanced > desktop.
You can find all kind of interesting settings by grepping output of gsettings
list-recursively.
