Etiquette for contacting potential advisors after submitting application
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...