Often when you get a young dog who got 'ditched' early in life you find a dog who has been deemed 'incorrigible' -- people got her, didn't train her, probably yelled a lot and gave up ... maybe more than once.
There is a good dog under all that -- but *you* folks have to be completely and utterly consistent. Out of that will come gold, but it will take a while.
First -- she's young, strong, energetic with WAY too much time on her paws and way way WAY too much freedom.
You can't just put the dog in the crate -- you have to TRAIN them to the crate. (Unless you want a dog that screams, cries and tries to dig and eat her way out of the crate!) You set that crate up as the BEST place o the planet -- feed her in there, give her treats in there and spend time in there.
This dog should not be 'loose' anywhere until you truly have a handle on the behavior. Even if you have to let her drag a leash in the house (and while I'm housetraining I leash them TO me so I know every move they make).
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And that's only the beginning -- training takes time. It's not 'convenient' ... but it's necessary.
I agree with everyone above -- she's not getting nearly enough exercise. When you get home someone needs to be playing, exercising and doing stuff WITH her virtually all the time. The time you spend now to train her will pay off in a dog that is responsible ... later. But you have to put in the time ... NOW.
Bottom line -- she's too untrained and too immature to make good "choices". She hasn't bonded well with her pack yet and you have to teach her that pleasing you is the most important thing on the Planet!
At night she shouldn't be able to just "let herself out". Doggie doors are not for everyone. Because rather than a dog simply chosing to go out and elminate and come in "like it should" it's simply a license to go out and DO STUFF BECAUSE I"M BORED. How do you stop it? You prevent it. Plain, pure and simple.
But then that means you have to fall over the dog and "deal" with it inside? Yep, it does. Your wife may not want to do that ... but frankly, everyone in the family has to be on board with this, but really often once someone begins to see the dog respond to this and they begin to get some positive feedback (rather than being annoyed with the darned thing all the time) then things begin to work.
My best suggestion would be you all need to find a good (not a chain store 20 minute one) but a GOOD obedience class to get into. And EVERYONE in the family needs to go. Take turns -- first you take the first set and work the dog, then let your wife do the next one. The beauty of obedience classes is that it's more than the dog who learns. It's as much for teaching the humans HOW to teach the dog than it is for the dog to learn.
Janet, above, mentions NILIF. Nothing in Life is Free - it's GREAT. But ... again the whole family has to be in on it but it will transform your life with THIS dog. But ... included in this means that the dog can't just let itself out whenever IT wants to. You control everything. Not convenient at first, but trust me -- they learn FAST. And ... so will the family.