By the power of truth, I, while living, have conquered the universe.
The rear aft of the Dawn contains the hangar, there's bound to be something like a SOEIV John can use for when Cortana wakes him.
On that subject, no John does not die. Cortana drops a beacon for the UNSC but she notes they won't be found for years (since they're likely still outside the Milky Way as Mendicant Bias sent them to the Forerunner world).