It is possible that $\sigma$-closed forcing destroys projective stationarity:
Suppose $\mathcal A$ is a maximal antichain in $\mathrm{NS}_{\omega_1}^+$. Feng-Jech have shown that
$$\mathcal S=\{N\in [H_{\omega_2}]^\omega\mid N\prec H_{\omega_2}\wedge \exists S\in\mathcal A\cap N\ \omega_1\cap N\in S\}$$
is projective stationary. Suppose $V[G]$ is a ($\omega_1$-preserving) generic extension of $V$ so that $\mathcal A$ is no longer maximal, i.e. there is a stationary $T$ with $S\cap T$ nonstationary for all $S\in\mathcal A$. I claim that $\mathcal S$ is no longer projective stationary in $V[G]$. Otherwise, we could find a countable $M\prec (H_{\omega_2})^{V[G]}$ with $T\in M$ so that for $N=M\cap (H_{\omega_2})^V$:
- $N\in \mathcal S$
- $\omega_1\cap N\in T$
By 1., there is $S\in\mathcal A\cap N$ so that $\omega_1\cap N\in S$. But $M$ knows that $S\cap T$ is nonstationary, so there is a club $C\in M$ with $C\cap S\cap T=\emptyset$, however $\omega_1\cap N\in C\cap S\cap T$, contradiction.
Now if $\mathrm{NS}_{\omega_1}$ is not saturated,such an extension can be realized by $\sigma$-closed forcing:
Let $\mathcal A$ be a maximal antichain in $\mathrm{NS}_{\omega_1}^+$ of size $>\omega_1$. The forcing $\mathbb P$ consists of conditions $(s, f)$ where
- $s\in 2^{{<}\omega_1}$
- $f:\mathcal A\rightarrow \omega_1$ is a partial function with countable domain
- For $\gamma \in \mathrm{dom}(s)$ and $\gamma\in T\in\mathrm{dom}(f)$: If $\gamma\geq f(T)$ then $s(\gamma)=0$.
with the obvious order. The idea is that the first components generically build a characteristic function for a new stationary set $T$ and the second component makes sure that $S\cap T$ is bounded for any $S\in\mathcal A$. $\mathbb P$ is clearly $\sigma$-closed. The only non-trivial thing to check is that $T$ is really stationary in the extension, and here is where we need that $\mathcal A$ is large. Suppose $p$ forces that $\dot C$ is a club. Build a continuous elementary chain $\langle X_i\mid i<\omega_1\rangle$ of countable elementary substructures of $H_\theta$, so that $X_0$ contains all the relevant information. As $\mathcal A$ is large, $\triangledown (\mathcal A\cap\bigcup_{i<\omega_1} X_i)$ is costationary. This allows us to find $\alpha=\omega_1^{X_\alpha}$ with
$$\alpha\notin\bigcup(\mathcal A\cap X_\alpha)$$
Now build a $X_\alpha$-generic sequence through $\mathbb P\cap X_\alpha$ starting with $p$. If $q=(s, f)$ is the limit of that sequence then $q\Vdash\check\alpha\in \dot C$ and $\mathrm{dom}(f)=\mathcal A\cap X_\alpha$. We can now extend $q$ to $q'=(s\cup\{(\alpha, 1)\}, f)$. This is a condition by our choice of $\alpha$ and $q'\Vdash \dot C\cap \dot T\neq\emptyset$ where $\dot T$ is the canonical name for $T$.
Regarding question 2, it is consistent (from a measurable) that there is a semiproper forcing that makes $([\omega_2]^\omega)^V$ even nonstationary. The forcing is adding a Cohen real and then shooting a club through the complement of $([\omega_2]^\omega)^V$. Cox-Sakai proved that the semiproperness of this forcing is equivalent to a form of the Strong Chang Conjecture. Any semiproper forcing changing the cofinality of $\omega_2$ to $\omega$ would do the trick as well, of course.
I do not whether or not proper forcing can destroy the projective stationarity of the full $[\kappa]^\omega$.
