You can relate $B_2$ balls to the domain of analyticity of your function $f$. For instance, if $d=1$, $f\in B_2(r,L)$ implies that $f$ is analytic on $\{z : \vert z\vert < \sqrt{e^r}\}$ and reciprocally, if $f$ is analytic on $\{z : \vert z\vert < \sqrt{e^\rho}\}$ then $f\in B_2(r,L)$ for all $r<\rho$. You should have similar relations in higher dimension.

To put it differently, if you can say something about the domain of analyticity of your function $f$ you can then use Cauchy's formula to control the decay of the coefficients $a_k$, and therefore you could directly get an answer to your Question 3.