Proof of a vector calculus identity

In https://arxiv.org/abs/hep-ph/0010057 the following vector calculus equality is claimed without proof although in note [4] the cryptic comment is made that "The relation is essentially the momentum space identity $(mathbf{k}timesmathbf{A})^2=mathbf{k}^2mathbf{A}^2-(mathbf{k}mathbf{A})^2$ in position space":

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Indeed there is the vector relation:
begin{align}
int mathbf{A}^2(x)d^3 x
& = frac{1}{4pi} int d^3 x d^3 x’ frac{[nabla times mathbf{A}(x)] cdot [nabla timesmathbf{A}(x’)]}{vert mathbf{x}-mathbf{x’} vert}
& qquad + frac{1}{4pi} int d^3 x d^3 x’ frac{[nabla cdot mathbf{A}(x)] [nabla cdot mathbf{A}(x’)]}{vert mathbf{x}-mathbf{x’} vert} tag{6}label{6}
& qquad + rm{surface terms}
end{align}
Each of the two terms is positive; hence (up to the surface term question) we can minimize the integral of $mathbf{A}^2$ by choosing $nabla cdot mathbf{A} = 0.$ With this choice the integral of $mathbf{A}^2$ is minimal in accord with our above remarks and is expressed only in terms of the magnetic field $nabla times mathbf{A}$

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This $eqref{6}$ is indeed a very interesting identity and Gubarev, et al, go on to show it also in relativistically invariant form.
When $mathbf{A}$ is the vector potential, $mathbf{B}=nablatimesmathbf{A}$, then in the Coulomb gauge $nablacdotmathbf{A}=0$ and
$$int mathbf{A}^2(x)d^3 x = frac{1}{4pi} int d^3 x d^3 x’ frac{mathbf{B}(x) cdot mathbf{B}(x’)}{vert mathbf{x}-mathbf{x’} vert} + rm{surface terms}$$
Ignoring the "surface terms" in the infinity and assuming that the integrals of $eqref{6}$ are positive indeed then we have the gauge independent minimum on the right side dependent only on the $mathbf{B}$ field:
$$int mathbf{A}^2(x)d^3 x ge frac{1}{4pi} int d^3 x d^3 x’ frac{mathbf{B}(x) cdot mathbf{B}(x’)}{vert mathbf{x}-mathbf{x’} vert}.$$
I have two questions:

1. I would like to see a more detailed explanation of the proof based on the momentum space – position space equality
2. Why is it obvious that on the right side of $eqref{6}$ each of the two integrals is positive?