In the notation of the question,
$$
\int f(x,y)\nu(y)dy=W_\sigma\ast \nu(x)
$$
where $W_\sigma(x)=e^{-|x|^2/2\sigma^2}$ and $\ast$ denotes convolution. Thus, if
$$
\int f(x,y)\nu(y)dy=\lambda\nu(x)
$$
then by taking Fourier transforms:
$$
\lambda \hat{\nu}= \widehat{\lambda \nu}
=\widehat{W_\sigma\ast \nu}
=\widehat{W_\sigma}\widehat{\nu}
=c_1W_{c_2}\hat{\nu}
$$
since the Fourier transform of a Gaussian is a Gaussian which is rescaled in mass and variance from the original one by some constants $c_1,c_2>0$ whose exact values are easily computable but unimportant. So evidently $\hat{\nu}$ must be zero in the complement of a sphere centered at $0$, since the spheres centered at $0$ are the sets on which the multiplier equality $\lambda=c_1W_{c_2}$ can hold.

Thus, the eigenfunctions will be inverse Fourier transforms of tempered distributions supported in spheres centered at the origin and there are lots of these - the ones that must be mentioned are the harmonic polynomials or more generally the harmonic functions of tempered growth, which correspond to the degenerate sphere with radius zero (the constants are included in these), other than this the most familiar ones are the Bessel functions, which correspond to uniform surface measure on a nondegenerate sphere. There are lots of references for this kind of thing, the one that comes to mind immediately is *Classical Fourier Analysis* by Loukas Grafakos.

Now as for $L^2$ eigenfunctions, there are none other than zero... because any such eigenfunction would have to be the inverse Fourier transform of an $L^2$ function which is supported in a sphere and since spheres have measure zero, $0\in L^2$ is the only such function.

Now for the second question, the operator
$$
\nu\mapsto \int f(\cdot, y)\nu(y) p(y)dy
$$
for a centered Gaussian $p$ is not translation invariant and so will not be given by a Fourier multiplier. I would guess that even in this case there can be no nonzero $L^2$ eigenfunctions but I can't think of a decent proof at the moment. However, I suspect that you may have intended to ask instead for eigenfunctions of the standard operator
$$
\nu\mapsto \int f(\cdot, y)\nu(y) dy
$$
which are in $L^2(pdx)$ - in that case there are lots of them. Any harmonic function of polynomial growth and the inverse Fourier transform of $f d\mu$ where $f$ is smooth and $\mu$ is uniform measure on a sphere centered at zero, for instance.