Chance-constrained is an optimization problem that ensures, probability of meeting a constraint is above a certain level. The formulation of this problem is generally defined as :
$ \min \mathbb E_\zeta[f(x,\zeta)] \tag{1}$ such that $$P(h(x, \zeta) \geq 0) \geq \alpha $$
Where $P$ is the probability function and $\alpha$ is a reliability index between $[0,1]$.
Q1) Is it possible to consider violation of an equality constraint in chance constrained problem as well? For instance, if $h(x,\zeta)\ge 0$ was an equality constraint, could we still formulate the chance constrained problem as :
$$ \min \mathbb E_\zeta[f(x,\zeta)] \quad\text{ such that } \quad P(h(, \zeta) = 0) \geq \alpha$$
Q2) It is often said in the literature by setting $\alpha=0.5$, the solution of the chance-constrained problem $(1)$ is equal to the deterministic representation of the problem. For example if you have $N$ scenarios $\zeta_{i=1:N}$ with probability of $\rho_{i=1:N}$, the solution of (1) for $\alpha=0.5$ is equal to solution of
$$\min \mathbb E_{\mu}[f(x,\mu)] \quad \text{ such that }\quad h(x, \mu) \geq 0 \quad \text{ where } \mu = \sum_{i=1}^N \zeta_i \rho_i$$
Is it a strict rule or just when stochastic process is defined by a normal distribution?
Q1) If $\zeta$ is distributed continuously, the probability that $h(x,\zeta)=0$ is zero if $h(x,\zeta)$ is strictly increasing or decreasing in $\zeta$ for all $x$. So it might be impossible to maintain the equality constraint, since it is measure zero that you succeed with a continuous space of outcomes. If $\zeta$ has finite support, you might be able to do this, but for similar reasons, there might be no solution.
Q2) This does not seem true unless $h(x,\zeta)$ is linear in $\zeta$, so that you can ram the expectation operator through. Otherwise, suppose $h(x,\zeta)$ is strictly convex in $\zeta$. Then $\mathbb{E}[h(x,\zeta)] > h(x,\mu)\ge 0$ implies $\mathbb{E}[h(x,\zeta)]>0$ by Jensen's inequality, and the non-linearity seems to matter for deciding whether or not you are close enough to satisfying the constraint. What literature is this?