Is there a contradiction between isotropy and the Big Bang? Disclaimer: I'm not asking whether the Big Bang happened at a point. I'm asking whether the fact that the universe is isotropic and that the Big Bang happened contradict each other.
To be honest I am just starting to learn General Relativity and cosmology but I feel like this question has a profound basis. If we assume that the universe "started" by the Big Bang (which I naively imagine as just the expansion of a sphere) then surely it must have started out from a certain point, which is still in our sphere universe. However we also know that the universe is isotropic, that there are no "special" points. How can these two facts coexist since if the Big Bang really did happen then there must be a characteristic point in our universe, a special point, which contradicts the principle of isotropy. So what am I missing, if I'm missing anything?
 A: Your assumption of a three dimensional sphere is wrong, because we are in the General relativity framework in cosmology, i.e. we have space time, and the Big Bang happened , in the original model, at a (0,0,0,0) four dimensional point. 
The three space  dimensions are like the surface of a sphere ,in  a two dimensional analogue, and time is at the radius of the analogue. This means that all space points now were at the origin of the Big Bang. 
The current Big Bang model posits a fuzzy beginning due to quantum mechanical uncertainties, in effective quantizations of gravity.


I'm asking whether the fact that the universe is isotropic and that the Big Bang happened contradict each other.

The quantum mechanical fuzziness and the inflation period in the plot generate the isotropy. Before introducing the inflation period there could be no thermodynamic equilibrium, due to the light cone separations at the early universe. 
Take a usual balloon and assume it started expanding from a point, where all the surface was concentrated at a point. All points at the surface of the balloon were at the origin where expansion started. This is an analogue of the BB  eartly universe in two space dimensions and one time.
If at the early universe there were no light cone then  yes, you would expect a thermodynamic equilibrium . But  it is General Relativity and four vectors that describe the universe, and at the very early times it is the  light cone that does not allow thermodynamic equilibrium, and the inflation period has been included in the BB model to fit the observed isotropy.
A: Your intuition is faulty because you are imagining universe’s expansion as similar to the expansion of gases from a bomb blast. Something like the “front of the universe” is expanding and occupying some sort of imaginary space, moving away from a central point. A wrong intuition that may come from the name “Big Bang”.
When we say the universe is expanding, we mean the distance between two adjacent points in universe is becoming bigger, just like a rubber band with 2 marks that is being stretched.
Imagine a container with a lot of balloons having a small amount of some chemical reactants whose reaction produces gases. After some time, the balloons will start expanding and occupying a bigger volume. However this expansion isn’t starting in a particular point, it is coming from everywhere since each of the balloons is pulling over the ones around it. This is the same type of expansion that happens when you boil a cup of rice.
For a better intuition, imagine the entire universe as a 4D Mobius “strip” balloon(1), inflating from a zero size volume.
A: Einstein assumed space was isotropic and homogeneous because he believed the universe was closed. 
And it simplified the model. 
Also, an isotropic and homogeneous space is needed for constant curvature. 
Cosmology principals assume the universe is isotropic and homogeneous - basically has a "stream line flow". 
In fact, the cosmological principle reduce a 10 parameter theory to a single function and discrete parameter which characterizes Robertson-Walker space time. 
However to determine the structure of the universe, we need an accurate measure of distance - which currently doesn't exist.
For instance, Google for the "Fingers of God" - which is a plot of all the measured red shifts in a quadrant of the sky - for instance the Virgo Galaxy. It looks like there's a big finger pointing back to the Earth. 
And another interesting plot is the "Pancakes of God" - which suggests the universe is just a web of Machian objects.
