Basically, the "current" facts cover most of the bases nicely enough.
1) "A star", that is, the Romulan sun, explodes. Spock doesn't say when in relation to his mission, but gives a year that may be 2387 or thereabouts. PIC gives a year that may be 2389 or thereabouts. Both are fine and compatible, because both only give the year in relative terms, not absolute.
2) The explosion threatens the galaxy. Well, everything relating to Romulus does. Plus, if something close to Romulus blows, it's likely to be bad news to everybody, including at least Klingons and Feds, because all these major players seem to reside close to each other.
3) There is time to react to the explosion: Spock builds a ship, Picard builds a fleet. The time comes before the actual explosion, and is of unknown length because supernovae aren't clockworks.
4) Multiple plans to help are in motion. At least two fail. Many more may succeed. In the end, plenty of Romulans are saved, but not a miner's wife.
5) Spock has made a promise to save Romulus, and intends to keep it. His controversial scheme involves red matter. It fails in saving Romulus, at least in part because Spock guessed the timetable wrong. It succeeds in another goal, though, even though Spock has "little time" to pull off that part.
6) Spock's red matter vial nips the supernova in the bud, well within the Romulan system, by consuming the expanding wave of destruction in an effect that propagates from the vial along the surface of the wave. Given the small distances involved, this is plausible, and moreover saves all other star systems (possibly including the nearby Vulcan and Earth) from destruction further down the road.
7) The mighty Romulan Star Empire is no more, without its capital system. However, not all Romulans are gone, nor are all Romulan worlds stated to be gone.
That's plenty of boxes checked there. What remains unknown gives a much shorter list:
1) Spock's plan for saving Romulus is not known in detail, or even in basic principle. Chiefly because it fails. But it involves red matter, which behaves differently in every single deployment seen in the movie. A behavior that would save Romulus may well exist, too.
2) We never learn why Spock needed a fast ship, or indeed a special ship at all. But perhaps he only needed a ship compatible with housing red matter, and chose the fastest one available because there were no other criteria that would take precedence.
3) We don't know the name of the star that blew. Was it perhaps Hobus? Nor do we get a complete rundown of planets lost or Romulans killed. But there's time for that yet.
Personally, I think PIC has swept all the problems under the rug competently enough. Nothing about Spock's story remains contradictory, merely mysterious. And it's solid canon that Vulcan mysticism is poorly comprehended.
Timo Saloniemi