Are US PhD admission offers binding?
As far as I know (at least here at Georgia Tech) all the conditions (if there are any) are included in your offer. I have never heard of us rescinding an offer after it has been accepted.  If there is a deadline, though, you must accept by then or the offer may no longer be valid.

That said, I think nobody would be offended if your asked, if it puts your mind at ease.  I would not mention the "getting good grades in your MS" or similar things.

Just send a note to whomever you have been in contact with and say something like "I'm very happy to have been accepted into the program, and am looking forward to coming.  I am pretty sure I've completed everything I need to do to accept admission, but I just wanted to check one last time. I have a number of other acceptances I'm about to decline and I just wanted to check that everything is in order with my admission before I decline the other offers.  Thanks for your help and understanding!"