In the Handbook of Set Theory, Foreman has (essentially) the following proposition (3.9):

Suppose $\kappa_n>...>\kappa_0$ and $\lambda_n>...>\lambda_0$ are regular cardinals and $(\kappa_n,...,\kappa_0) \twoheadrightarrow (\lambda_n,...,\lambda_0)$. If either GCH holds or there are countably many cardinals between $\lambda_0$ and $\kappa_n$, then for all structures $\frak A$ on $\kappa_n$, there is $\frak B \prec \frak A$ such that $| \frak B \cap \kappa_i | = \lambda_i$ for all $i \leq n$, and $\lambda_0 \subseteq \frak B$.

Question 1: Can we drop the assumptions of GCH or countably many cardinals between the top and bottom cardinal?

Question 2: The argument I know using GCH also uses the regularity of the $\kappa_i$'s and $\lambda_i$'s. But it is superfluous in the case of countably many cardinals between. So can we drop the regularity assumption in the GCH case?

In the Handbook of Set Theory, Foreman has (essentially) the following proposition (3.9):

Suppose $\kappa_n>...>\kappa_0$ and $\lambda_n>...>\lambda_0$ are regular cardinals and $(\kappa_n,...,\kappa_0) \twoheadrightarrow (\lambda_n,...,\lambda_0)$. If either GCH holds or there are countably many cardinals between $\lambda_0$ and $\kappa_n$, then for all structures $\frak A$ on $\kappa_n$, there is $\frak B \prec \frak A$ such that $| \frak B \cap \kappa_i | = \lambda_i$ for all $i \leq n$, and $\lambda_0 \subseteq \frak B$.

Question 1: Can we drop the assumptions of GCH or countably many cardinals between the top and bottom cardinal?

Question 2: The argument I know using GCH also uses the regularity of the $\kappa_i$'s. But it is superfluous in the case of countably many cardinals between. So can we drop the regularity assumption in the GCH case?

Question 3: The stronger Chang conjecture can be shown indestructible under $\lambda_0^+$-c.c. forcing. Does $\lambda_0^+$-c.c. forcing preserve the ordinary Chang conjecture? What about forcing of size $\lambda_0$?