2 I corrected the erroneous $I(\mu,\nu)$ to $I(\mu-\nu)$ and added ", finite or not,".
Concerning the first question: We have $I(\mu,\nu)>0$ I(\mu-\nu)>0$whenever$I(\mu,\nu)$I(\mu-\nu)$ is defined, finite or not, and $\mu$,$\nu$ are different signed Radon measures with equal total masses (or rather charges). This, with any finite dimensional Hilbert space in place of the plane, is Example 3.3 in http://www.ams.org/journals/tran/1997-349-08/S0002-9947-97-01966-1/home.html , where the assumption $\sigma\neq0$ is missing.
Concerning the first question: We have $I(\mu,\nu)>0$ whenever $I(\mu,\nu)$ is defined and $\mu$,$\nu$ are different signed Radon measures with equal total masses (or rather charges). This, with any finite dimensional Hilbert space in place of the plane, is Example 3.3 in http://www.ams.org/journals/tran/1997-349-08/S0002-9947-97-01966-1/home.html , where the assumption $\sigma\neq0$ is missing.