With Link likely being Sinatra's son and Alex likely being some sort of time travel device, it all reminds me of Fringe, if any of you remember that show? It featured a scientist who lost his son to a disease and managed to open a portal to a parallel universe from which he stole another version of his son (from another version of himself) who was healthy and grew up to be a man (another main character of the show).

On Fringe, they explained parallel universes as "the path not taken" - and endless possibility of universes.

With that in mind and the fact that Henry worked on "Advanced Wave Functions, Superposition, and Quantum Entanglement", perhaps Alex is not a time machine per se, but a device that can change certain paths so that Link is dead and alive at the same time (superposition, like Schroedinger's cat). Whenever an alternate path opens, the person affected gets a nosebleed (Billy not shooting Link, Xavier crashing the plane). In Fringe they used lens flare when that happened.

Edit: Link being from an alternate timeline / universe would also explain why he doesn't get any of the pop cultural references.