Fridge Logic: The Messiah has an Orion Propulsion System and was assembled in space. It's way too small to be one. (Orion was a military project to build a deep space craft in the late 50's, it works by exploding nuclear bombs behind a very thick metal plate and getting hit by the explosion. Most designs are assembled on the Earth and are about 1 mile in Radius).
Further, if it was an Orion Craft, which could achieve the course required by the ship to not only reach the comet, but match course with it, there would be a heck of a lot more nuclear weapons on the ship. No reason to not simply try again at altering the course of the comet fragments.
The Messiah was a big ship, as it needed to carry everything the astronauts would need for what's essentially a deep-space journey of roughly one year. To that end (under the heading of Crazy-Prepared) why not load several cruise missiles adapted for spaceflight and destroy the comet by using visual guidance to shatter it far out in space by hitting it with a couple to create fissures and then flying a nuke inside? Better yet, why not just tell the world, use Earth's combined resources to build a fleet of Messiah-class ships to hit the comet as mentioned above, or (just as good) have the fleet soft-land one of the vessels, lock it on the surface, and use the ship's Orion Drive to divert it by fractionally adjusting its course away from Earth?
Of course, you want to make sure that the astronauts give themselves enough distance from the comet before they detonate the nukes.
If you're going to soft-land engines on the comet to make it steerable, then Orions probably aren't the best choice; running them would be like repeatedly hitting the comet with a very large hammer. A better choice would be a steam rocket: you don't need to lift any reaction mass because you'd just use the local landscape.
Also, why doesn't the Messiah have at least one Soyuz capsule as an escape vehicle? Not only would they have been useful (only one person would have been required to make a Heroic Sacrifice), but they could have jury-rigged the radio equipment inside those capsules to re-establish contact with Earth.
Fridge Horror: If no one over the age of fifty would be selected, then the sheer number of older U.S. citizens who were either euthanized (to spare them from suffering through the horrors of the comet strikes) or given the means to commit suicide would have made the Holocaust look like a small-town Santa Claus parade.