On Kirk's death: From what I could tell, there was no indication of how much time passed between "death" and Tribble-eureka. Presumably they'd have taken Kirk out of that compartment and directly to sick bay; it might have been a matter of a few minutes. Even with 2013 medicine, people can be dead for several minutes (even an hour or two if they pass out in a frozen lake or something) and still be revived. It doesn't seem like a huge stretch to imagine that superblood could stave off acute radiation poisoning just long enough to get Kirk to a future-hospital. You don't really have to assume that the blood was capable of fully regenerating a decaying dead body.

On Starfleet headquarters: are you sure that was Starfleet headquarters? The computer stated that because of the malfunctions (or whatever) the target could not be guaranteed. And the ship crashed initially into the ocean, which probably wasn't Khan's intention; that says to me that maybe he missed. Unless I'm forgetting some piece of evidence, which is certainly possible. Anyway it obviously destroyed some buildings and stuff, but I don't remember anything about them being Starfleet buildings in particular.