Oh I know how you feel. When the weather got nicer here, i've been letting my guys on the balcony during the day to get some fresh air. I live on a second floor, so they are confined to the balcony with the stairs blocked off. After when they are all shuttled inside, Mooky is the only one who will meow at the back door to go back out. Sometimes he will do this around 11:00 p.m.
First I tell him firmly NO and if he continues, then he gets a squirt with the water bottle and he stops.

To answer some of your questions:
Yes I would continue to let her out on her leash. Maybe you can get a stake that you can stick firmly in the ground and tie a good rope or a very light chain, that can attach to her leash and let her roam around a bit.
In regards to her rushing the door, keep a water bottle close by so that before you open the door, just point it at her and tell her firmly, NO. If she attempts to go towards the door, give her a squirt.
She will, hopefully, get the message as Mooky has after a few squirts from the bottle. Now I just have to point it at him and he stops his meowing.
Hope this helps you.