Is there an example of how SOS can be used to show infeasibility of a set of multivariable equations? Lets say one is given a set of $m$ real polynomial equations in $n$ variables, $P_1 = P_2 = P_2 .. = P_m =0$. I understand that there is some theorem which says that if there is no solution to these simultaneous equations then one necessarily will be able to find polynomials $Q_1,Q_2,..,Q_m, R_1, R_2,..,R_k$ such that $deg(P_iQ_i)$ and $deg(R_i^2)$ are all  $\leq d$ for some $k$ and $d$ such that,  $\sum_{i=1}^m Q_iP_i + \sum_{j=1}^k R_j^2 = -1$


Can someone kindly reference this theorem and its proof? 


Now lets say one is looking at a $d$ which is not sufficiently large. Then one will be looking at the set of all polynomials of the form $S = \sum_{i=1}^m Q_iP_i + \sum_{j=1}^k R_j^2$ such that $deg(P_iQ_i)$ and $deg(R_i^2)$ are all  $\leq d$ for various $k$. But this is a convex set in the set of polynomials? And $-1 \not \in S$. Then there will exist a separating hyperplane between S and $-1$ and the normal to that plane gives a degree-d pseudo-distribution on S. 


Does someone have a concrete example of the above phenomenon? 
Why is this pseudo-distribution indistinguishable from an actual distribution on $S$ if one is only looking at moments $\leq d$?
The theorem you are referring to is known as the Positivstellensatz, but you are citing it incorrectly; given only polynomial equations, the relevant theorem would be the Nullstellensatz. See also lecture notes of Laurent and blog posts of Amir Ali Ahmadi.

Regarding pseudodistributions, Barak defines them as linear operators for low-degree monomials which satisfy the crucial property $E[P^2] \geq 0$ for low-degree $P$. These are all the properties you need when looking at low-degree moments.