A counter-example to this "intuition" that $\pi(\theta_0)>0$ is necessary, when $\pi(\cdot)$ is the prior density and $\theta_0$ is the "true" value of the parameter is the [minimaxity result of Casella and Strawderman][1] (1981): when estimating a Normal mean $\mu$ based on a single observation $x$ with the additional constraint that $|\mu|<\rho$, if $\rho$ is small enough, $\rho\le 1.0567$, the minimax estimator corresponds to a uniform prior on $\{-\rho,\rho\}$, meaning that $\pi$ gives equal weight to $-\rho$ and $\rho$ (and none to any other value of the mean $\mu$). 

Similarly, when considering admissible estimators, Bayes estimators associated with a proper prior on a compact set are usually admissible, although they have a restricted support.

In both cases, the frequentist notion (minimaxity, admissibility) is defined over the possible range of parameters rather that at the "true" value of the parameter (which brings an answer to Question 4.)


  [1]: https://projecteuclid.org/euclid.aos/1176345527