The second problem can be solved using the following Liouville type theorem, which was first used in works related to De Giorgi conjecture (see for example L. Ambrosio and X. Cabré). Let $u$ be a positive solution of $\Delta u=gu$ and $v$ a bounded solution of $\Delta v=fv+gv$. Then consdier $\psi=v/u$. It satisfies
Using $\psi\eta^2$ as a test function with $\eta\in C_0^\infty$, we get
$$\int fu^2\psi^2\eta^2+u^2|\nabla\psi|^2\eta^2\leq\int v^2|\nabla\eta|^2.$$
So if $v$ is bounded, you can use a classical technique by choosing a good test function $\eta_R$ with some $\log$ dependence to show the RHS goes to $0$ as the radius $R\to\infty$.
Note that this is something special in dimension $2$. The existence of positive solution $u$ is also related to the stablity of $\Delta-g$.