If I pause the execution of the world, will Greenfoot.getKey() still record the last key that was pressed, even when the world is paused? If not, how could I achieve the execution of the world to be paused whilst still running the getKey() method?
No. That is not possible. Although, you could create a public method that will except a keystroke and store it for future use, if that is what you want.
Why would you want to 'pause' the execution of the scenario and still allow keystrokes to be directed to it? Maybe(?) you did not explain what you wanted clearly enough and there IS(?) a way to do what you want.