Etiquette for contacting potential advisors after submitting application 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?
So, questions about their current papers etc are not the same as asking about your admission progress.

Yes, ask the about the papers and projects but not about your admission status.

You should address questions about your admission status to the admissions department and, they may have already published a link, or email, for such contact.

As for mentioning other offers, if you are border-line case for them and they know you have other offers that may help their decision...