Objects don't outright disappear: As with other event horizons, they freeze in time and become redshifted as they approach the cosmological one (they also become fainter due to an increase in proper distance).

The cosmological event horizon is the light cone at future infinity and the asymptotic boundary of the observable universe: Light emitted within the horizon will take a finite time to reach us, whereas light emitted right at the horizon would take an infinte amount of time to arrive; in a way, light emitted beyond the horizon still moves towards us in the sense that the comoving distance decreases, but we'd have to wait a longer-than-infinte amount of time for it to arrive...

Because they are beginning to move faster than c, when these objects are at the very edge, due to the space between us and the edge expanding faster than c. This is called the cosmic horizon of the universe - sort of like an inside out event horizon of a black hole. A particle inside the horizon with a speed of c can still reach us (albeit very strongly redshifted), a particle outside the horizon would have to travel faster than c to reach us. The thing is, our own universe is still not old enough, that it can have a proper event horizon. It is estimeted, according to a model that assumes dark energy is a cosmological constant, that the universe must be at least 16GY old for a cosmic horizon to develop.

Oh, and by the way, an expansion faster than the speed of light is consistent with relativity. Special relativity only describes local laws of physics -- you and the edge are not "local". And general relativity doesn't have a constraint on a maximum velocity between 2 arbitrary points in spacetime.

The expansion is FTL.