"Why'd they make Neil a cop?"
He mentions that psychic ability could be brought on by trauma. So it was probably meant to be implied that something in his law enforcement/armed services career caused him the trauma that made him psychic.
"Where were they?"
I'm assuming it's some sort of sanctuary for people with psychic abilities. I dont think that they wanted to send her away I think they just wanted to be left alone but Neil agreed to talk to her
The one guy said "close the damn door this ain't no freak show." So I think that they just thought she wouldnt understand them.

I was wondering that myself. I was also confused why they had to make Neil a cop. It feels like the filmmakers had to cut out some scenes but it made parts of the movie confusing.