I think you are starting from square one - as if you had a tiny puppy. She probably should not have free reign of the house until you can get her housebroken.
I understand your reluctance to use the crate - but it's still better than having to rehome her if you aren't able to get her under control. I also think xpen would be a good idea if you are totally opposed to the crate.
When she does go outside (I'm assuming you are training her outside) - PRAISE HER like crazy! Have a big old praise party and some REALLY GOOD treats ~ the NANOSECOND she pees or poops. She'll catch on quickly.
I have a nice blanket and toys inside Lucy's crate and a blanket over top of it (like a den) and she loves to sleep in there - so maybe it won't be so bad.
Regarding a dog that runs outside - my Westie will - everytime. I learned the concept of training him to stay out the entire area surrounding the door (that's MY space) by watching Cesar Milan. I don't use all his methods - but physically keeping Ringo out of that space by using my body - seemed to work well.
Lastly, just to get started - you can try "Nothing in Life for Free", which is a very humane way of gently reinfocing your position as leader. Your dog is probably skittish and frightened (due to his prior life) and a strong leader my help him calm down.
Nothing in Life is Free Gaining control of your dog humanely
Keep us posted! You can do it with patience and consistency.