I have applied to a bunch of PhD programs in computer science but have not heard back from several of them, and my top choices are among those schools. I am thinking about contacting a few professors at those institutions about their progress in the graduate admissions process, as I already have other questions that I want to ask them (questions about their papers, current projects, etc.)
Is it OK if I ask them about the status of my application and/or the admissions timeline after I ask my real question? Will the answer differ if I do/don't know them in person? Or is it best for me to just refrain from asking such questions in any case?
In addition, if I mention offers that I have already received in the email, will that be considered rude?