Is there an example of how SOS can be used to show infeasibility of a set of multivariable equations?
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.