I could not find any instructions on moving apps from the Desktop
into sub-Folders. In case anyone else is interested, here is
how I ended up doing it. Sorry if I miss a step.
In this example I will move the Calculator app into a folder called
"Etc".

mkdir Desktop/Etc bin

cat >> .local/share/applications/mimeapps.list << EOF
[Default Applications]
application/x-desktop=run-desktop.desktop
EOF

cat > .local/share/applications/Etc.desktop << EOF
[Desktop Entry]
Name=Etc
Exec=nautilus /home/furios/Desktop/Etc
Type=Application
Icon=folder
Categories=Utility;
EOF

cat > .local/share/applications/run-desktop.desktop << EOF
[Desktop Entry]
Version=1.0
Name=run-desktop
# Exec=/usr/bin/dex %U
Exec=/home/furios/bin/deskopen %U
MimeType=application/x-desktop
Terminal=false
Type=Application
NotShowIn=GNOME;
EOF

As root, edit /usr/share/applications/org.gnome.Nautilus.desktop and
append this to the end of the "MimeType=..." line: ;application/x-desktop

As root, mv /usr/share/applications/org.gnome.Calculator.desktop Desktop/Etc

NOTE making *.desktop files executable may be a security risk!

cat > bin/deskopen << EOF
#!/bin/sh
Self=`basename $0`
dir=`dirname $1`
arg="$1"
sus=0
log="$HOME/.local/log"
if [ ! -d "$log" ]; then mkdir -p "$log"; fi
log="$log/$Self"

block=`echo "$arg" | sed -e 's/[[:alnum:]\.\/_-]//g'`
if [ x"$block" != x ]; then sus=1; fi
if [ x"$dir" != "x/home/furios/Desktop/Etc" -a \
x"$dir" != "x/home/furios/Desktop/System" ]; then sus=$((sus+2)); fi
if [ ! -r /"$arg" ]; then sus=$((sus+4)); fi

if [ 0 -lt $sus ]; then
echo `date +"%b %e %R:%S"` `hostname`: "$sus: $arg" >> $log
exit 1
else
/usr/bin/dex "$arg"
fi
exit 0
EOF

chmod 755 bin/deskopen

At this point you should have an Etc folder on your Desktop containing
the org.gnome.Calculator.desktop app. It should run when tapped, although
the on-screen keyboard may come up. I am not sure why it does, but you
can dismiss it by holding down the white "______" bar at the bottom of
the screen. I suppose you could rename "org.gnome.Calculator.desktop"
to "Calculator.desktop" if you like. Now to change the icon...

grep -i icon= Desktop/Etc/org.gnome.Calculator.desktop
(shows Icon=org.gnome.Calculator)

find /usr/share/icons | grep org.gnome.Calculator
(shows /usr/share/icons/hicolor/scalable/apps/org.gnome.Calculator.svg)

app="/home/furios/Desktop/Etc/org.gnome.Calculator.desktop"
gio set $app metadata::custom-icon \
file:///usr/share/icons/hicolor/scalable/apps/org.gnome.Calculator.svg

Don't forget the "file://" part!