You can go until you run out of gas (longer if you refill w/out shutting off the car). There isn't a range. The car will go until you shut it off. After that though, you'll need a key to start it again.
Keep in mind though, that part of Infiniti Connection is a service that will track your stolen vehicle. They'll be able to tell you (or the police) where the car is. They can't remotely shut it off, but if it is shut off, they can remotely lock it.
Car doesn't have a "valet key" per se. It does have the metal blade key inside your keyfob. You could use that to lock/unlock. And the idea of locking the car while it is running if you're leaving it parked isn't a bad idea. I think you'd have to do that with the button inside the door - the one outside on the handle, nor the keyfob button should work while the car is running.