Inquiring about PhD status and mentioning standing offers
Is it OK to send an email asking about application status and explaining that I have other standing offers which I have to make decisions about quickly?


Yes, it is more than okay.  If you think you might prefer school X over your current list of accepted schools, definitely let them know.  At worst they will be indifferent to this information.  At best, they will be very grateful to know this and be given the opportunity to enroll you.  They may for instance fly you in on short notice to interview / recruit you.  In no case (well, no reasonable case...) will they respond negatively.


  Would this have any negative impact on my application? 


As mentioned above: absolutely not.  Rather it could have a positive impact, including in some cases resulting in better funding for you.

Added: The above is general advice without factoring in the thought that school X may be better than the schools to which you've already been accepted.  The advice is still valid, but the response from a top school may be less dramatic: top programs tend to make their admissions decisions more independently and tend to have good funding anyway.  (Nevertheless they would be happy to know that you are still interested in them before you make your decision.) A (fairly long) while back an old friendly acquaintance of mine was visiting the Harvard math department as a prospective student after already having been admitted to MIT (which is not a measurably worse department, I should mention).  He expressed to me the hope that MIT's interest would engender a "bidding war" between the two schools.  This did not happen and in fact the student did not stay in any graduate program for very long.