As the universe ages, will we see more stars or less? After a very long time will we see more stars (due to the fact that more light is get to us) or less stars (as the universe expends and light have to pass larger distance)?
In general, can stellar objects go outside of the scope of the observable universe or is it only growing with time? Calculations are always better than just discussion.
Does it have any connection to dark matter?
 A: Let's forget about the lifetime of stars and their formation rates, which means the idealized question I'm going to answer is closer to the question

Will we see more galaxies as the universe ages?

A galaxy is visible if it has emitted photons that reach us, ie if our past lightcone intersects its worldline.
Assuming FLRW cosmology (see this answer for some maths and a nice picture), I'd emphasize the following facts:


*

*galaxies that were visible in the past will remain visible in the future

*we will keep seeing new galaxies being born


This means that indeed the number of visible galaxies increases.
However, it's also true that


*

*the difference in age between a given galaxy as we see it and our own will increase and there's a limiting age beyond which we'll never be able to see

*intensities will drop and redshifts increase, making galaxies harder and harder to detect


This means that there are galaxies we currently see as they were, but we'll never be able to see as they are right now and future scientist might no longer be able to see other galaxies (not gravitationally bound to us) for technological reasons.
A: This is a rather lengthy answer as I tried to go a bit in depth; there is a short summary at the end.
Will we see more or fewer stars with time?
The short answer to this is: We see less stars with time, due to the fact that cosmic expansion is accelerating. Although what we really see at the relevant distances are galaxies; single stars are far too far away to be resolved.
The first thing to realise to understand this is that the speed limit of special relativity doesn't apply to cosmic expansion. If you imagine the galaxies as raisins in an enormous, rising dough, what SR tells you is that nothing can move through the dough faster than light. But if expansion is the same everywhere, and the dough is large enough, then an arbitrarily small speed at which the dough would rise would make all raisins beyond a certain distance recede from your local raisin faster than light. This "certain distance", in the Universe, is called the Hubble Distance. Galaxies farther away from the Hubble Distance are receding from us faster than light.
This, however, is not the limit beyond which light cannot reach us. If a galaxy is receding from us faster than light, a photon emitted from said galaxy towards us will initially be receding from us, but it will crawl to regions of space - at local light speed - which are receding from us a bit slower than the one from which it was emitted, and so it will recede a bit slower from us than before, etc., until it finally enters a region of space that is receding from us slower than light, from which point it is trivial that it will eventually reach us.
This is not always the case, though. There is, in certain cosmologies (including our own with an accelerating expansion), a distance,from beyond which a photon emitted will never reach us. This happens if the Universe expands faster than the photons can traverse it. This distance is called the cosmic Event Horizon, and while it expands in terms of absolute distances, it is shrinking in terms of co-moving distance. The co-moving distance is the distance measured in a coordinate system that expands along with the Universe - such a system will have galaxies more or less sitting in the same coordinates throughout the history of the Universe, and so the Event Horizon having a shrinking co-moving radius from us directly tells us that fewer objects will be inside this radius in the future.
The way we can in theory observe these distant galaxies leave our event horizon - and the reason why it has its name - will be very much like we would observe an object falling into the event horizon of a Black Hole: relativistic effects will redshift the light emitted from such an object indefinitely, which also slows down the time as observed by us. So the object will appear to slow down asymptotically to a complete freeze, while the light will be redshifted towards infinity and become undetectable by us, although you could argue that it will never, in theory, disappear completely. The last we will ever see of such a galaxy is a point in its history (that is not special to the galaxy itself), after which we will never receive more recent information about the galaxy - this is why it is called an event horizon.
Can stellar objects go outside of the scope of the observable universe or is it only growing with time?
This is not easy to answer with a clear "yes" or "no", because the term "the observable Universe" is not clearly defined.
The event horizon that I mention above is defined as:

Event Horizon: The distance, from beyond which a photon emitted towards us now will never reach us.

And we just established that this distance is shrinking in co-moving coordinates, so yes, galaxies are escaping our cosmic event horizon as we speak.
But we can see that the Event Horizon has the property that is is a cut-off in time, as well as Space: For any galaxy, it is the last event in its history that we will be able to observe. But we could also ask a different question:

What is the farthest distance, at which we can see the first event in an object's history?

Or in other word, how far can a photon emitted at the time of the Big Bang, have travelled today? At which distance are we observing the Big Bang? We call this distance the Particle Horizon:

Particle Horizon: The distance from which a photon emitted during the Big Bang will reach us now.

We normally use the Cosmic Microwave Background as a proxy for this; it currently has a redshift of ~1100 and a proper distance of 46 billion light years from us (remember, that is the current size of that space, but it has expanded continuously while the photon was travelling, so the photon will still have travelled only the 13.8 billion light years we would expect from the age of the Universe). While the event horizon is growing in absolute, but shrinking in co-moving coordinates, the Particle Horizon is growing in both coordinate systems, so that horizon is expanding forever.
Summary
To answer this question properly, we have to think in terms of both time and space. On one hand; for all galaxies, there is a point in their history that is the last we will ever observe, so in that sense, galaxies (and thus stars) are receding out of the "observable Universe" as we speak.
On the other hand, the distance at which we can see the first events in the Universe is expanding and always will be. But the time limit on how much of the history of these new regions we can study before they slow down to a freeze and redshift to infinity will get ever shorter.

PS: The math behind
There is an article that is relatively easy to read for someone with a background in physics, written by someone way smarter than me, on ArXiv.org, explaining this whole thing in better theoretical detail.
A: Star formation will slowly start to decrease in galaxies as the universe ages because of the conversion of gas, such as hydrogen, into heavier elements, such as carbon and iron. Essentially the universe is slowly running out of fuel. Stars have a hard time fusing heavier elements.
Eventually there will be no stars left. What will remain are black dwarves, black holes, neutron stars, etc. These will slowly decay over time into a cosmic soup of fundamental sub-atomic particles.
The expansion of space will mostly make it harder to see galaxies not stars.
A: This is a really interesting question. I think the easy answer has already been given:


*

*Stars require lighter elements (Hydrogen, Helium) in order to
generate energy (and hence light) from fusion. There is a finite (but
thankfully enormous) amount of these lighter elements in our
universe, all (in a "black boxed" sense) of which is gradually
heading down the energy slope towards being fused into iron. Eventually
this fuel will run out, everything will cool down, and no new visible light will be generated.


Many stars now, no stars future, requires a decline. QED.
But the substance of your question was I think more about objects in the visible universe, ie objects from which we can possibly obtain any information. This one is much harder to answer, I think the real answer is "we cant be sure" the timescales the universe operates on so far exceed the period of our observation that we must hesitate to state anything with certainty. 
That said, our best shot at giving any answer is to exrapolate from those limited observations. They are all we have. I am going to choose to proceed on the assumption that "the universe tomorrow will behave exactly the same as the universe today" and "the universe at distance operates the same as in our locality" i.e. that the laws of physics are both space and time invariant.
The facts we know: 


*

*the universe is expanding. Now by this we dont simply mean that the
objects in the universe are getting further apart, although they are,
we mean that the rate at which they move apart is increasing, and not
only is it increasing, it is increasing proportional to an objects
distance from the observer. This is analagous to some underlying stretch in the space between objects.

*The rate at which this expansion or stretch is occuring is accelerating.

*Light travels at a finite speed. ~3*10^8m/s in a vacum. This is the fastest speed that any information can travel in the universe.


From these three statements you would conclde that the visible universe must be shrinking. There would be (an unimaginably huge) distance apart two objects could be at which the acceleration due to distance would be ~3*10^8 m/s/s and within the second objects in that part of the universe would be lost to us
A key point here however is that relativaty tells us that no two objects can exceed the speed of light relative to one another.
This means that no matter how hard you push, whatever force comes from this "stretching" cannot possibly force objects out of the visible universe. 
From this I conclude that in fact the scope of the visible universe in terms of objects from which we can obtain information, is probably increasing, and it will probably continue to increase for the entire future of the universe. Further I contend that this increase in scope is likely an asymtotic aproach to some maximum value.
Hope this helps.
A: I red some times ago about a scenario in which the number of visible objects is becoming smaller and smaller. This is basically due to the Hubble's law: the further two objects are, the faster they move away from each other and when the speed exceed the speed of light, no news can come from them any more. If you take into account that the expansion of the universe is accelerating (the Hubble constant is increasing) this becomes even worse.
I don't like to think that a lot of things in the universe are constantly dropping outside my light cone, however this is what our current observations suggest and is probably still better that to have the universe to compress and finally squeeze on itself.
A: Will we see more stars?
The big bang blasted matter in all directions, resulting in the particles moving away from each other. These slowed down due to gravitation pull on each other. First forming small clumps due to their proximity, later with the clumps of clumps pulling together. This process keeps going recursively eventually forming stars, planets, and other cosmic bodies, grouping together to form solar systems, galaxies, etc.
The stars eventually die when their fuel run out, collapsing in on their own weight, with the more massive ones forming black holes. Eventually over time everything starts pulling closer to each other due to the gravity between them, the solar systems are pulled into the center of their galaxies, the galaxies pull on each other and combine to form bigger galaxies, and even the black holes and massive black holes at the center of galaxies swallow each other to form even bigger black holes. In the end the entire universe collapses in on itself, perhaps resulting in another big bang, with the whole process starting all over again.
So the bigger question to your question is rather to which stage in this life-cycle of universal evolution are you referring to? We could say that at the moment of the big bang there were no stars because massive volumes of gas have not yet have the chance to form, nevertheless had the chance to start to compact to build up pressure and form stars due to their size and gravity.
Or are you referring to a later stage in this life-cycle, when most of the matter were still moving away from each other, forming starts and galaxies along the way, during this time there will be an increase in the number of stars formed.
Or are you referring to when everything collapses in on each other, when there will be less stars formed, with supernova explosions along the way and new gas clouds being formed, which may eventually create new stars, but are ultimately also swallowed by other stars and black holes, eventually resulting in fewer stars.
Can stellar objects go outside the scope of the observable universe?
The observable universe is only a tiny speck compared the the actual size of the universe. With "us" as the observer, we can only observe as much as our technology allows, there will always be something further that we cannot get to though observation, for example, has any of our technology been able to look though a black hole so see what is on the other side? With refraction and gravity, light and energy can slightly bend around objects, but we cannot see what is exactly closely behind the black hole. And in most cases we cannot observe everything in the spectrum of waves, for example to see what's on the other side of a dark cloud of matter, you would rather resort to far-infrared imaging.
Calculations?
Being a maths and physics nut myself, I just crave any opportunity to scribble formulas down and work stuff out, but in this case you would have to be more specific of what you want to calculate, and "when" this calculation should be performed.
Number of Stars at Start of Time = 0
Number of Stars at this moment = sum of stars in all solar systems in all galaxies
Basically it's uncountable since we cannot count, or even approximate, what we have no idea of what the boundaries are.
A: If the age of the universe is $1 / H_0 $, then as the universe gets older $ H_0 $ would get smaller.
So theoretically the recessional velocity of distant galaxies gets smaller and you should see more.
Of course observation falsifies $ H_0 $ getting smaller and thus the expansion models are falsified.
Meaning the universe looks roughly the same at all times, whether a trillion years ago or a trillion years from now. 
A: I am probably sure that you have heard about how the sun is going to become a red giant over the next few billion years and later become a incredibly dense white dwarf.This concept applies for all stars of all the universe, not that they all become white dwarfs, but that they all run out of fuel at some point move into another star phase.
There are mainly 2 star death outcomes. They either become a white dwarf and fade, or they die explosively in a supernovae leaving behind either a black star or a black hole. Although the decayed elements from these 2 outcomes will form new stars in nebulas. The production of black holes out weighs the formation of new stars. 
Therefore trillions of years from now, the universe will be a sea of black holes. A few trillions years later, the universe will simply contain 1 supermassive black that is inconcievably big compared to nowaday black holes. But due to Hawking Radiation this black hole will fade and the universe will ultimately be nothingness.
Although this theory is favored in the scientific community, different outcomes are also possible, like another Big bang.
