Cosmic event horizon changes over time The cosmic event horizon is the comoving distance beyond which the light signal from distant galaxy can never reach us. With dark matter, the event horizon converges to a finite value so does it mean that it is constant over time? i.e. If we can observe a galaxy now, we can always observe it in future because the comoving distance is unchanged over time and will be always within the event horizon?
 A: 
With dark energy, the event horizon converges to a finite value so does it mean that it is constant over time?

The event horizon will converge with the hubble radius in about 16 billion years:

It will approach, but never reach, a fixed value of about 18 Gigalightyears.

If we can observe a galaxy now, we can always observe it in future because the comoving distance is unchanged over time and will be always within the event horizon?

As you can see in my animation above the scale factor will expand rapidly  while the event horizon stays fixed. But the particle horizon grows even faster than the scale factor. That means that galaxies we can observe today will also be visible in the distant future since they stay within the particle horizon, but as soon as they moved out of our event horizon it will no longer be possible to send an actual signal to them, or to receive a signal sent by them after they left our event horizon.
So yes, we will always see old images of this galaxies as they were before they fled our event horizon, but we will never see what happened to them afterwards. Their signal will be redshifted to an infinite duration.
The same plot in comoving distances shows that the event horizon is shrinking compared to the coordinates, while the particle horizon (and future light cone) approaches, but never reaches, a fixed comoving coordinate (see Link).
A: In a universe dominated by dark energy, the universe would expand forever. The equation for the comoving cosmological horizon in a universe that will expand forever is given by:
$$d(t)=\int_t^\infty\frac{dt'}{a(t')}$$
Since we assume in the most accepted model that dark energy is due to a cosmological constant, the scale factor as a function of time in a universe with only dark energy would be approximately $$a(t)\propto e^{Ht}$$ where $H$ would be the Hubble parameter and constant.
Thus, we find that the cosmological at any time $t$ in a dark energy dominated universe is:
$$d(t)\propto\frac{1}{a(t)H}=\frac{1}{\dot a(t)}=\frac{e^{-Ht}}{H}$$
In an expanding universe, this result is decidedly not constant (it definitely decreases), so the comoving distance is not unchanged and comoving objects do not remain in the observable universe (remember that at $t\to\infty$, the past light cone corresponding with what we can observe then would asymptotically overlap the event horizon, so comoving coordinates crossing outside the event horizon also cross outside of the observable universe).
The proper distance to the cosmological horizon in such a universe is constant. That means one could theoretically travel to the limits of it (limits here meaning arbitrarily close to but not at it) and then travel back in finite time, but the expansion would carry non-bound, comoving objects outside this limit eventually.
