You are responsible for your dog. Your dog is no one else's responsibility.
Hope I am wrong. But, I do not think this will end well for a child and/or the dog.
You are the owner. You are the one that will be sued if your dog bites someone. If the dog destroys property. You are the one that will be sued.
A lot of landlords will not allow dogs that have a history of biting people. Your animal could be hurt or worse. Not to mention what could happen to a child. Your are trying to place responsibility on others. It is not their legal responsibility. They do not own the dog.
You are at the mercy of irresponsible people and there's not really anything you can do to change their behavior. Did you ever consider that their behavior is part of a 'grand plan' to get the dog or you out of the house? If you can't or won't prevent them physically from letting your dog out then there's a good chance that down the road there will be a serious issue that will come back to bite You!
this is what happens when you deal with roommates, search in the forum for 'my roommate' and read through the horror stories of dealing with roommates. Most times the only solution is to move away from them and be done with it.
You can't fix Stupid!
"So basically, if my room mate brings a kid in to the house, leaves the dog gate open, gives the kid food and leaves the kid unsupervised with my dog, and my dog bites the kid, and even though my room mate knowingly bypasses the preventative measures, knowing the dogs behavior, and knowing what has happened before, and doesn’t tell us ahead of time she’s having the kid over to visit, and my dog bites the kids finger trying to snatch a chicken nugget, I am 100% liabile and they can sue me? And they get to keep being stupid irresponsible dumbasses and receive no consequences for their actions?"
Stop being the idiot here. Either buy a big, strong crate for the dog (and stop providing excuses as to why the dog can't use one), or get rid of the dog or the roommates.