What strengthenings of measurability does the Mostowski collapse of the ultrapowers possess?

Here's what I mean. Let $j:V\rightarrow M$ be an elementary embedding with critical point $\kappa$ where $P(M)$ holds (given some third order predicate). Let $MO$ be the Mostowski collapse of $Ult_U(V)$ for some nonprinciple $\kappa$-complete ultrafilter $U$. When does $P(MO)$ also hold?

A perfect example is measurability itself. $j$ has critical point $\kappa$. Letting $P(M)$ be "there is an elementary embedding $j_0:V\rightarrow M$ with critical point $\kappa$", it can be seen that $M$ holds iff $MO$ holds.

But does it hold for the following strengthenings of measurability?

• Let $\kappa$ be $\theta$-strong. Is $V_\theta\subset MO$?
• Let $\kappa$ be $\theta$-supercompact. Is $MO^\theta\subset MO$?
• Let $\kappa$ be $n$-superstrong, and $j_0:V\rightarrow MO$ be the composition of the mostowski collapse function (between the ultrapower and $MO$) $\pi$ to the canonical ultrapower embedding. Is $V_{j_0^n(\kappa)}\subset MO$?
• Let $\kappa$ be $n$-huge, and $j_0:V\rightarrow MO$ be the composition of the mostowski collapse function (between the ultrapower and $MO$) $\pi$ to the canonical ultrapower embedding. Is $MO^{j_0^n(\kappa)}\subset MO$?