i think the answer is yes.

let M=M_1. let lambda=omega_2^M and say G collapses omega_2 to omega_1. let f: omega_1-> lambda be an increasing cofinal function. let construct a continuous chain (M_i: i<omega_1) of submodels of M|lambda such that f(i) is in M_i.

let N_i be the transitive collapse of M_i. then N_i is an initial segment of M, this follows from comparison.

let then M_omega_1 be the direct limit of N_i. there is a map pi: M_omega_1-> M_i|lambda. this map has the range of f in its range and its critical point is bigger than omega_1, so it is identity.

you can do this with any gamma<omega_3^M. fix f: omega_1-> lambda and g: omega_1-> gamma that are increasing and cofinal. then construct a chain that exhaust f and g. similar argument would show that M_omega_1 is M|gamma.

i think the answer is yes.

let $M=M_1$. let $\lambda=\omega_2^M$ and say $G$ collapses $\omega_2$ to $\omega_1$. let $f\colon\omega_1\to\lambda$ be an increasing cofinal function. let construct a continuous chain $(M_i: i<\omega_1)$ of submodels of $M|\lambda$ such that $f(i)$ is in $M_i$.

let $N_i$ be the transitive collapse of $M_i$. then $N_i$ is an initial segment of $M$, this follows from comparison.

let then $M_{\omega_1}$ be the direct limit of $N_i$. there is a map $\pi\colon M_{\omega_1}\to M_i|\lambda$. this map has the range of $f$ in its range and its critical point is bigger than $\omega_1$, so it is identity.

you can do this with any $\gamma<\omega_3^M$. fix $f\colon\omega_1\to\lambda$ and $g\colon\omega_1\to\gamma$ that are increasing and cofinal. then construct a chain that exhaust $f$ and $g$. similar argument would show that $M_{\omega_1}$ is $M|\gamma$.