Well, when you say "stays," does this mean she lives there, or just visits occasionally? The Court will generally only order such "background" checks of people who have significant caretaking responsibilities with the child. Moreover, the Court will not generally order drug testing unless there's an allegation of prior drug use. In any event, you're encouraged to schedule a consultation with a NYC Child Custody lawyer.
* If you found my answer to be "HELPFUL," or the "BEST ANSWER," please feel free to mark it accordingly.
If there is an issue of drug usage, then a request is appropriate. If it is just a desire to know, them it is not appropriate. Consult with a local family law attorney. Custody cases are difficult to handke on a pro se basis.
If this answer is helpful, then please mark the helpful button. If this is the best answer, then please indicate it. Thanks. For further information you should see an attorney and discuss the matter completely. If you are in the New York City area, then you can reach me during normal business hours at 718 329 9500 or www.mynewyorkcitylawyer.com.