Could the spatially flat universe start small? The global space has been measured flat within a small margin of error. According to FLRW, the flat universe has always been infinite. At the time of the Big Bang the universe was infinitely large with the infinite energy density at every "point". This means that at the beginning any finite "volume", no matter how small, had an infinite total energy.
I understand that "volume" and other descriptive properties cannot be applied directly to the singularity. What I am actually referring to is limits. E.g., the total volume of the observable universe becomes arbitrary small, as we trace it back in time arbitrary close to time zero:
$$ \lim_{t\to 0}{V}=0 \tag{1} $$
Obviously, the total energy of the observable universe is not infinite and has never been infinite in its lifetime. This means that the observable universe stated from an infinitely small "volume" (as described above), essentially from a "point".
Please note that this description is different from the naive view that "the Big Bang happened at a point", as explained here:
Did the Big Bang happen at a point?
While the Big Bang did not happen at a point, out observable universe indeed started from a "point" defined by $(1)$ above.
I realize that the content of the observable universe changes in time with the space expansion. This however is irrelevant to my question. The only relevant condition is that the energy of the observable universe is always finite, but obviously not constant.
If the observable universe started from a "point" (as defined) in an infinitely large "space", then any other "point" in this "space" is not in our past light cone, is causally disconnected from our observable universe, and cannot influence us in any way other than by contributing to the global space curvature being flat.
If this is correct, then there seems to be no tangible difference between the universe starting infinitely large or infinitely small. If our observable universe started from a "point" in an infinitely large "space" and any other "point" is causally disconnected from us, then why do we need to consider these other "points" as "existing" in the first place? What would stop us from simply postulating that the entire universe started flat, but small, while initially coinciding with the observable universe? 
Is there anything wrong with this line of thinking? Thanks for your expert insight!
 A: 
If this is correct, then there seems to be no tangible difference between the universe starting infinitely large or infinitely small

If you assume the universe is infinite, it has to be infinite at any given time. and only at the initial time, there is big bang singularity. 
If the universe is infinite it was always infinite. At $t=10^{-10000000}$ it was still infinite. It has to be geometrically. But at $t=0$, we have a singularity.
You wrote that infinitely small means like a point, But the universe cannot be squeezed into a point, as you know. So even it's infinitely small it's still infinite. 

If our observable universe started from a "point" in an infinitely large "space" and any other "point" is causally disconnected from us, then why do we need to consider these other "points" as "existing" in the first place?

Because the universe has to be infinite at any given time. So these points exist by mathematical definition. 

What would stop us from simply postulating that the entire universe started flat, but small, while initially coinciding with the observable universe?

The universe started from a singularity. If its flat, it has to be infinite again. No matter how small it is.
If you mean "Why we cannot think our universe started like as an observable universe" my answer would be this. 
1-Universe is the thing that encounters everything. So it still has to start from a singularity. And if its flat it has to be infinite
2- CMBR radiation shows that there is no preferred direction in the universe. Which points out that there cannot be any -away from point type- expansion. So even the observable universe seems to start from a point, It actually did not start from a point. Observable universe has no "real" center, There is just the universe and we have a limit on what we can see.  
A: When you write "point" in scare quotes, what you're essentially doing is reinventing the notion of boundary constructions. A couple of good surveys on this topic are:
Sanchez, "Causal boundaries and holography on wave type spacetimes," http://arxiv.org/abs/0812.0243
Ashley, "Singularity theorems and the abstract boundary construction," https://digitalcollections.anu.edu.au/handle/1885/46055
The main thing to realize about boundary constructions in GR is that attempts to apply them to general spacetimes have failed. They are very convenient in the context of Penrose diagrams, but we don't have a useful general theory of them.
Your point about the nonfalsifiability of the existence of unobservable regions of spacetime is fine, but it has nothing to do with cosmology. You can take Minkowski space and do silly things like removing a point from it, or removing everything except for a certain region. This has no consequences for an observer whose past light cone avoids the missing points, but it's a silly thing to do, and we have no laws of physics that would help us to decide what the removed parts of the spacetime should be. This is why relativists only usually want to discuss maximal extensions of spacetimes.
A: 
Could the spatially flat universe start small?

As you are mentioning the observational data we have indicate spatial flatness of the universe. Given this is true there are two possibilities regarding its shape. 


*

*the shape is like a plane. Then the universe was infinite at the big bang

*the shape is like 3-Torus. Then the universe is finite and was finite at the big bang.  Indeed the WMAP data of the CMB seemed to show a signature suggesting this possibility. That wasn't confirmed however by the Plank mission. 


In both cases the observable universe was finite at the big bang.
The question is what do we mean if we say big bang? The hot and dense state of the early universe, often called Planck era, or the state before the "slow-roll inflation" started? I tend to prefer the former because the latter is under investigation and not well understood.  

What would stop us from simply postulating that the entire universe started flat, but small, while initially coinciding with the observable universe?

Well the flat 3-Torus is still not ruled out even though the CMB doesn't reveal a signature. If the shape of our universe is a 3-Torus which is much much larger than our observable universe then we can't expect such a signature. So it seems we will never know for sure. Cosmologists mainly believe that the universe is infinite because the 3-Torus is a non-trivial solution.
But in case the universe is a 3-torus then the observable universe would be a tiny fraction of it at the big bang (as interpreted above). 
A: A point could not remain a point while retaining some dimension larger than another, although a sphere or cube of nearly infinite smallness might be indistinguishable from a point, without magnification that might require an inaccessible amount of energy.
I believe that some curvature of spacetime is essential for any temporal repetition, including the operation of clocks. One side of any inflexible object following a curved path necessarily covers more distance in the same amount of time than the side closer to the focus of the curve, and curved objects are more adaptable to the wobbling or precession required for such motion to proceed smoothly, so that symmetrically curved objects are much more common in nature than symmetrically rectangular solids, which are usually rendered asymmetric rather quickly by collisions and friction.  (No inflexibility can suffice to prevent the distortion of any object by Lorentz contraction during its acceleration to relativistic speeds, although the passage of light around it will cause that distortion to appear as a change in the object's scale or angle of approach, rather than a change in its shape.)  Relativity is formulated to reflect natural formations, rather than artificial ones, and the cubic torus analogized to the pac-man game in some literary representations of spacetime is usually represented schematically as the inner surface of an hourglass or doughnut hole, even though the volume it contains may have been mathematically "cubed".  
Although the current CMB data shows space to be very nearly flat, its complete flatness would interfere with such interchangeability of space and time as our recognition that we do not see stars as they are now, but more-or-less as those same stars were when whatever light from them that we are seeing left them.  
I say "more or less" because even that "special relativistic" effect would not provide for gravitational distortion of the light rays, which might only be taken into account through General Relativity, and would be further complicated by the possibility that some of the objects causing such distortions (such as dark energy, dark matter, or black holes formed by the gravitational collapse of non-binary stars) might themselves remain invisible to us.  (The elliptical orbits of the more common stars whose binary partners would've become black holes would indicate the presence of those BHs to us.)    
