I'm guessing that dogs and pets are not allowed at your apartment complex, so you want to try to classify your sister's condition as a "disability" to circumvent the ban?
Based on what you described, I doubt it'd work.
First, you'd need to prove that your sister has an actual disability. While some of the behavior may be out of character for your sister, I don't think it reaches the level of being an actual "disability," such as blindness. At the very least, you'd need some sort of professional examination that can actually attribute your sister's behavior to the traumatic events in her life, as opposed to being just a teenager. You can try to claim she has PTSD (
http://www.ncbi.nlm.nih.gov/pubmedhealth/PMH0001923/), but it depends on how stringent the government official is. He may be sympathetic, he may ask for more evidence of actual disturbing behavior.
Second, you'd need to show that getting a dog is a legitimate means of addressing her condition, as opposed to something that "you think may work." What if it doesn't? Do you give the dog back? That sucks for the dog.
Based on your story, and assuming you didn't leave anything relevant out, if I were the government official, I'd tell you "Don't pee on my leg and tell me its raining." So no, I don't think you could get a dog under the FHA.