Which command can I use to minimize the current terminal window. i.e. this should do the same as a click on the minimize window control:

Which command can I use to minimize the current terminal window. i.e. this should do the same as a click on the minimize window control:

xdotool lets you programatically (or manually) simulate keyboard input and mouse activity, move and resize windows, etc. It does this using X11's XTEST extension and other Xlib functions.
You can minimize the current active terminal window like that
xdotool windowminimize $(xdotool getactivewindow)
Be aware: This is not specific to gnome-terminal
A few terminal emulators (not including gnome-terminal) support the escape sequence ESC [ 2 t to minimize the window:
echo -ne '\e[2t'