Let me turn my comment into an answer (to Q3, hence also for Q1).
I claim that for a fixed $f\in C_c(\mathbb{C})$, the map $$ \text{prob}(\mathbb{C}) \ni \mu \mapsto \int_{\mathbb{C}} f(z)d\mu - \lim_{r\to 0} \frac{1}{\pi r^2} \int_{\mathbb{C}} \mu_z([0,r)f(z) dz $$ is total-valuation-norm continuous, respects convex combinations and it vanishes on the extreme points (that is, $\delta$-measures). Therefore, it is identically zero.
It follows that for a given $\mu$, for every $f$, $$ \int_{\mathbb{C}} f(z)d\mu=\lim_{r\to 0} \frac{1}{\pi r^2} \int_{\mathbb{C}} \mu_z([0,r)f(z) dz. $$
As for Q2, I am not sure what do you expect. One way to answer would be using the inversion formula given above, that is given the collection $(\mu'_z)_z$ you can define $\mu$ using the above formula and then define the collection $(\mu_z)_z$ as you did. The initial collection is "consistent" iff you got back where you've started.