This is very interesting stuff. Your equation for time dilation is, of course, correct.
It seems to me that if you were a photon traveling at the speed of light then time would not elapse at all. Thus you, the photon, would be able to move around the universe “instantly”.
Is that correct?
Yes, that is true. You can even say that it hadn't even moved, since its travel distance would have contracted to zero by virtue of its traveling at light speed. Weird stuff, and just the tip of the cosmological iceberg.
Yes, that is correct. If a photon had consciousness, it would perceive the travel to occur instantaneously. We, as outside observers, would still see that it took a fixed amount of time to travel the distance, though.