The other answers seems to answer most of your questions, but I think one confusion remains: The speed of light as a maximum speed in the Universe (which is not the case).
First off, redshift doesn't go to infinity for objects receding at $v = c$. We easily see galaxies recede at superluminal velocities. In fact, this is the case for all galaxies with a cosmological redshift of $\gtrsim\!1.5$.
To understand how this is possible, imagine a photon emitted from the galaxy GN-z11 in our direction when the Universe was 400 Myr (million years) old, and the distance to that galaxy was $d_\mathrm{then}=800\,\mathrm{Mpc}$ (roughly 2.7 billion lightyears). At this time, the scale factor (the size of the Universe relative to today) was $a=0.08$, and the Hubble parameter was $H_\mathrm{then} \sim 1600\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, so the recession velocity of GN-z11 was
$$
v_\mathrm{then} = H_\mathrm{then} d_\mathrm{then} = 1\,300\,000\,\mathrm{km}\,\mathrm{s}^{-1} \simeq 4.3c.
$$
In the beginning, although the photon departed from GN-z11 at $v=c$ as it should, and even though it traveled in our direction, it was carried away from us by the expansion of the Universe faster than this, and thus receded from us at $v=4.3c-c=3.3c$.
However, expansion is ubiquitous, and thus also helped carrying the photon away at an increasing speed from GN-z11. At some point (when the Universe was 3.7 Gyr old), expansion had helped it reach the midpoint between GN-z11 and the Milky Way, and although its local velocity was always $c$, for a brief moment it stood still wrt. both MW and GN-z11. Then it began to increase its velocity, until it hit the Hubble Space Telescope last year with a speed of $c$.
Today, the distance of GN-z11 has increased to $d_\mathrm{now} = 9.9\,\mathrm{Gpc}$, and it thus recedes from us at
$$
v_\mathrm{now} = H_\mathrm{now} d_\mathrm{now} = 670\,000\,\mathrm{km}\,\mathrm{s}^{-1} \simeq 2.2c,
$$
which is still superluminal. However, due to the accelerated expansion of the Universe, photons emitted from GN-z11 today will never reach us.
Edit: Clearing up a few misunderstandings
the cosmological horizon is slowly getting closer to us
In fact, the opposite is the case: The particle horizon, which is what you link to, is defined as the farthest we can see, which is given by the distance that light has had to travel since the Big Bang. Since time increases, this distance always increases.
there are distant galaxies moving away from us at or faster than the speed of light.
This is true, as described above.
the speed of these galaxies is purely in reference to where you are measuring from
Again true. While we measure the recession speed of GN-z11 to be $2.2c$, an observer midway between us and them would measure half this speed.
the perceived time distortion is an illusion caused by our relative speed, time is not really grinding to a halt for distant galaxies
Time does indeed run slower in distant galaxies, as seen from us. But it does not come to a halt at $v=c$. In fact, time is simply dilated by a factor $(1+z) = 1/a$. That is, time in GN-z11, which lies at redshift $z=11.1$, runs slower by a factor $12.1$.
However, this is not an illusion, but a real effect. This is the very essence of relativity; you must accept that time (and space and simultaneity, etc.) is relative for different observers in order for physics to make sense. An example of when this time dilation is important is the time it takes for the brightness of distant supernovae (SNe) to decrease. All type 1a SNe have the same explosion mechanism, and increase and decrease in brightness in the same way (modulo a known effect but forget about that for now). But distant SNe are seen to decrease more slowly by a factor that exactly corresponds to the redshift they're at (plus one).
if you were in one of them, light would still move the same speed in relation to that location.
Indeed true.
I've been informed in class that I was mistaken, and no object can go the speed of light
Again, no object or information can travel through space faster than light. But space itself can expand at any rate.
