If the universe is expanding faster and faster, won't we reach a moment where the general relativity will make the mass so big, that the universe will have to decrease its acceleration?
The expansion of the universe is stranger than it appears at first glance. For a start, the Big Bang didn't happen at a point like some kind of explosion. When we say the universe is expanding we mean that the distance between any two stationary points (I'll get on what we mean by stationary in a moment) increases with time.
Suppose we take two points $A$ and $B$. For convenience we'll put $A$ at the origin then we can write the position (in space) of $B$ as $(x, y, z)$. Here the $x$, $y$ and $z$ coordinates are comoving coordinates, and when we say $A$ and $B$ are stationary we mean their comoving coordinates do not change with time. If you remember Pythagoras' theorem then you'll remember that the distance $d$ between $A$ and $B$ is:
$$ d^2 = x^2 + y^2 + z^2 $$
(This only applies to a flat universe, but our universe appears to be flat so that's OK)
When we say the universe is expanding we mean that the distance is actually given by:
$$ d^2 = a^2(t)(x^2 + y^2 + z^2) $$
where $a(t)$ is a function called the scale factor that changes with time. If $a$ is increasing with time that means the distance between $A$ and $B$ is increasing with time and that means the universe is expanding i.e. the distances between stationary objects are increasing with time. We generally take the value of $a$ to be one right now, so $a$ was smaller than one in the past and will be bigger than one in the future.
If you're interested I talk about how we calculate $a$ in How does the Hubble parameter change with the age of the universe?.
But to get back to your question:
I would guess that when you say mass increases you mean the fact that relativistic mass increases as you approach the speed of light. So if distant galaxies are speeding away from us their mass must be increasing. However those distance galaxies are (approximately) stationary with respect to us just like $A$ and $B$ in my discussion above. All that is happening is that the distance between us and the distant galaxies is increasing with time. So the distant galaxies aren't getting more massive, and for completeness they aren't suffering any relativitic time dilation either.
But there is a sense in which you are correct that mass could stop the expansion, though in our universe it doesn't appear this will happen. The way the scale factor changes with time depends on the average density of the universe. If the density is lower or equal to the critical density then $a$ just increases smoothly with time and the universe keeps getting bigger. If the average density is higher than the critical density then $a$ increases at first, but then peaks and starts to decrease again, which means the expansion of the universe slows to a stop then reverses.
As far as we can tell the density of our universe is too low for the expansion to reverse. In fact the presence of dark energy means the expansion will keep getting faster and in the far future the scale factor will increase exponentially with time.
Because relativistic mass only arises because of motion through space, not motion with space. For more details google "peculiar velocity" and "recessional velocity". Susskind explains this here.
Per dark energy hypothesis -
It does not increase mass or kinetic energy, because it is expansion of space itself, and universe is expanding with space, not into it.
Actually, it is supposed to be other way, expansion increases the dark energy in the universe, which is expected to be cause of the acceleration to begin with.
In one speculation, the process is expected to continue till everything flies apart including atoms and subatomic particles.
Not my opinion, it is per the hypothesis.
However, there are other speculations that the dark energy may change from repulsion to attraction some time in the future, and then, yes, the expansion would slow down.
They have already detected at least one cycle of accelerated expansion, and slowdown in the past. Last switch from slow down to acceleration is supposed to have happened nearly 5 billion years ago.