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The faster you move through space, the slower you move through time - the calculations show that as you approach lightspeed, time comes to a standstill.
How does time work beyond the cosmic event horizon? Space is stretched in every direction, meaning the further away something is, the faster it moves away from us.
But what about when that speed is more than $3 \cdot 10^8 \frac{m}{s}$ due to the distance away from us it is? What happens then?
First, let's clear up a few misconceptions:
The speed of light as an upper limit is valid in special relativity (SR). In general relativity (GR), which must be used to describe the expansion of the Universe, although locally (i.e. where SR is a good approximation) you cannot exceed the speed of light, there is no limit to the relative velocity of two objects.
By Hubble's law ($v=H_0 d$), all galaxies that are more distant from us than the "edge" of the so-called Hubble sphere $d_\mathrm{H} = c/H_0 \simeq 14.4\,\mathrm{Glyr}$ (billion light-years) recede from us faster than $c$. This is no hindrance for us to observe them; we regularly observe galaxies much farther away (see e.g. this and this answer).
The cosmic event horizon (EH) is the largest distance a galaxy can be at, and still emit a light signal that may reach us in a finite time. It is currently roughly 16 Glyr away, i.e. a bit farther than the Hubble horizon. We also regularly observe galaxies farther away than the EH, but we see them in the past (as we see everything else), and the light they emit today will never reach us (not because space expands, but because the expansion rate accelerates).
The particle horizon (PH) is the largest distance from which light has had the time to reach us since the Big Bang. Due to the expansion of the Universe, this is currently 46.3 Glyr away. Galaxies at this distance recede from us at 3.2 times the speed of light. We don't see any galaxies here because we look back in time to just after the Big Bang where they hadn't yet formed (and because the Universe at that time was foggy), but the most distant galaxy yet observed, GN-z11, is 32.2 Glyr away and recedes at $2.2c$.
However, you're right that the galaxies' recession from us makes their time go slower, as seen from us. Not by the SR factor $(1 - v^2/c^2)^{-1/2}$, but by the GR factor $(1+z)$, where $z$ is their observed redshift.
Galaxies receding from us at $v=c$ have a redshift of $z\simeq1.5$, so when we observe physical processes in such a galaxy — e.g. the decline time for the brightness of a supernova — it's slower by a factor $\simeq2.5$. GN-z11 is at redshift $z=11$, so things there happen slower by a factor 12.
The farther you look, the slower things happen, and if we could look arbitrarily far, you could in principle see the event right after the Big Bang in slow motion. You cannot see "zero o'clock", which would have an infinite redshift. And you can never, not even in theory, see anything that's farther away than it's light has had the time to travel, so not even in theory does it make sense to try to assign a time dilation factor to these events.
As far as we know, there are galaxies outside the PH. As time goes, the distance to the PH increases, both because space expands, and because light fro more distant region will reach us. But as soon as that light is able to reach us, the object that emitted it is inside the PH, and thus has a finite time dilation factor.
But remember that the time dilation is only from our point of view. If you could magically teleport yourself there right now, time would pass normally for you there. So nothing particular happens there.
Smarter guys will get through the complicated, maths-heavy explanations, but in term of wrapping your mind around the idea:
In that local frame of reference over the cosmic event horizon, you're not moving, no more than you're currently moving at higher speed than 3x10^8m/s from a over-the-horizon-distant point in space.
Keep in mind there is no such thing as a center in the universe, a bit like there is no center on the surface of a sphere.
Your question inherently assumes as objective, static entity, "the space". It is because on the Earth, the everydays experience is that we can stay where we are, or we can move somewhere, with a speed. But it is because we live on the Earth, trapped to its surface by its gravity.
Thus, we have an objective frame of reference, it is the Earth, and the speed of the things is automatically interpreted in this frame of reference.
Physics doesn't work so. There is no such thing than objective frame of reference. It is well known far before Einstein (Galilei understood it first, in the XVII. century). You can't say
"$\rm{X}$ goes with speed $\rm{v}$, then its time will be slower with $\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$".
Instead,
"$\rm{X}$ goes with speed $\rm{v}$ in our frame of reference, then its time will be slower with $\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ in our frame of reference".
Stars right beside our cosmological horizon stay in their frame of reference. Thus, they experience absolutely nothing special. For them, they are static and we are flying away with near-c or even with superluminal speed.
The situation becomes a little bit more complex by the expension of the Universe. This expansion is not like an explosion, i.e. not things are moving so faster, as they are more far away from some explosion center. Instead, the geometry of the spacetime is changing, distances becoming bigger and bigger with time. The Universe does not expand from some "center", because it has no center. More can you read about it in the Question Did the Big Bang happen at a point?. It is like as if we would live on a continuously stretching rubber sheet.
This type of expansion is not limited by $c$, because there are no velocities involved.
Thus, the stars out of our cosmological horizon can "move away from us" with superluminal speed. It doesn't mean that they would be superluminal. It only means that while their light could reach our telescopes, the Universe expands more. Thus, we can't ever see them. It is actually more: we have no way to interact with them in any sense. Effectively, they are not part of our observable Universe.
Thus, we have no way to know, what is there. If a big wall of bricks would exist just out of our cosmological horizon, we would not see it. However, there is a strong assumption that there are stars and galaxies also there. It is simply because there is no reason to think, that anything would be significantly different only because we can't see them.
