I have received an email from a US university asking if I am still available to accept an admission offer to their PhD math program. They mentioned that I was in the "top group" on their waitlist and they might have a spot open up in the next few days.
My understanding is that things move rather quicky after the April 15 deadline and hence I would not be given a lot of time to make a decision. My question is that would it be appropriate for me to contact the remaining 6 schools from which I have not heard yet to know the status of my application, and my position on their waitlist, if any, before receiving an offer from this school? I feel like I might not have enough time to wait for their responses after an offer is made. Do I need to include any details about the offer I'm expecting in my email to these universities? Similarly, is it okay for me to contact their current grad students pre-emptively to gauge their level of satisfaction with their program to help me decide? Roughly how much time is an applicant allowed to make a decision for offers made after the April 15 deadline?