Shortcut to open Terminal and run some commands

First you need a script like one from the following example:

#!/bin/bash

commands () {
    cd / # change directory, in this case to `/`
    echo "your_password" | sudo -S mkdir -p "test" # execute a command as root for which the password is automatically filled
    $SHELL # keep the terminal open after the previous commands are executed
}

export -f commands

gnome-terminal -e "bash -c 'commands'"

Don't forget to make the script executable!

Then, if you are not familiar about how to execute a script using a desktop shortcut, see Execute sh script from *.desktop file?.