How should I start a new iTerm terminal with shortcut?
You can change the Action in your Automator-defined service to an AppleScript action:


Select Library/Run AppleScript in the Actions window
Add the following AppleScript in the created Action and save:


if application "iTerm" is not running then
    activate application "iTerm"
else
    tell application "iTerm"
        create window with default profile
        activate
    end tell
end if


This opens iTerm if it is not running, else it opens a new window. See iTerm documentation for more examples of AppleScript scripting for iTerm.