Believe me, I understand. Let me know how it goes, as it is one I have been considering for when I can upgrade in the fall. Hopefully they will be caught up by then.
As for when to expect the phone to go dead, I would expect it to be the ship date (June 4 in your case). If it doesn't drop by then, they may ship it without a number and then transfer your number over the phone (must be from a different phone) after you get it. I really don't know how they do so through the mail. What I do know is they can not port the number while it is in service. Behind the scenes, they turn off your current phone, port the number, register it to their network, then assign it to the phone via the ESN number.