Did really everything begin with a state with very low entropy? As emphasized by Penrose many years ago, cosmology can only make sense if the world
started in a state of exceptionally low entropy. The low entropy starting point is the
ultimate reason that the universe has an arrow of time, without which the second law
would not make sense. However, there is no universally accepted explanation of how the
universe got into such a special state. Are there some observations that would really tell us that the early universe was with small entropy? Is this claim really consistent with our theories?
 A: Entropy is known to be strictly increasing (in the precise sense of a positive local energy production) due to the many dissipative processes in Nature. This is probably the most thoroughly verified fact in physics.
As a consequence, the total entropy of the universe (if this term can indeed be well-defined, which is somewhat questionable) must have been much lower in the past, as in an isolated system (and the universe is by definition isolated), the total entropy increases, too. 
This is independent of but consistent with current cosmological models. 
On the other hand, the question why this is so is difficult to answer, 
Possibly the quesion is moot, as the total entropy in the universe could also be infinite, in which case it was always infinite.
A: For an enormously detailed but still very readable answer to this question, see Sean Carroll's book "From Eternity to Here".  It's all about this subject.  Carroll also has a couple of lectures up at ted.com that give the highlights.
As we look out into space, we're effectively looking back in time.  When we see something 10 billion light years away, we're seeing light that left it 10 billion years ago.  So by looking at objects different distances away, we can see how the universe has changed with time.  There's also the cosmic microwave background radiation, which gives us the most direct information about the state of the very early universe.
A: The 2nd thermodynamic law is invalid when we add gravitation to an homogeneous infinite (*) ensemble of particles at rest (with temperature 0ºK).  
Why ?
What would you expect to happen if we add a small perturbation (a QM consequence) ?  
Think of a regular crystal where all particles are equally spaced and then one particle moves away from its initial position forming a hole.
A hole will expand because all particles in the exterior are less attracted to the center of the hole then before.
The temperature WILL GROW in the mass shell exterior to the hole and the hole will grow, in an accelerated fashion.  
In the real universe such holes are called VOIDS and galaxies are formed in the intersection of the voids.  
In my answer to PSE-anti-gravity-in-an-infinite-lattice-of-point-masses I show the equations of the gravitational field and graphics, under this scenario.

I got downvoted there, and I expect the same now, without argumentation, typical of believers that accept no evidence on contrary to their beliefs.  
(*) because the gravitational field grows at c speed it can be only 'large enough' .
