Collapsing the cardinals between two singular cardinals 
Question 1: Is it consistent that there is a forcing notion collapsing $\aleph_{\omega\cdot 2}$ to $\aleph_\omega$ without collapsing $\aleph_\omega$ or $\aleph_{\omega\cdot 2 + 1}$? 

If the answer to question 1 is affirmative, it will probably involve a forcing construction. It is natural to ask whether a similar situation can occur "naturally":

Question 2: Does some large cardinals assumption imply that there is a pair of singular cardinals $\mu < \lambda$ and a forcing notion collapsing $\lambda$ to $\mu$ without collapsing $\mu$ or $\lambda^+$? 

 A: For question 2; let $\kappa$ be a supercompact cardinal, and consider the diagonal Prikry forcing $\mathbb{P}$ to change the cofinality of $\kappa$ to $\omega$ and collapses all cardinals in $(\kappa^+, \kappa^{+\omega})$ into $\kappa$ and preserving all other cardinals. Let $V[G]$ be the resulting extension. But note that then there is some intermediate submodel $V[H], V \subset V[H] \subset V[G],$ which is essentially the ordinary Prikry extension of $V$ for changing the cofinality of $\kappa$ and preserving all cardinals. 
Now $V[G]$ considered as a generic extension of $V[H]$ has the required property. 
For question 1 add collapses to make $\kappa=\aleph_\omega,$ and define $V[H]$ similarly so that the cardinal structure of $V[H]$ and $V[G]$ below $\kappa$ is the same.
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It is possible to get a stronger result. Start with a supercompact cardinal and force with supercompact Radin forcing with a suitable measure sequences over $P_\kappa(\kappa^{+\kappa}).$ The forcing preserves $\kappa$ inaccessible, adds a club $C$ of former regulars into $\kappa,$ and for any $\alpha$, a limit point of $C$, it collapses all cardinals in $(\alpha, \alpha^{+\alpha}]$ into $\alpha$ but preserves all other cardinals below $\kappa.$ Call the extension $V[G].$ We can arrange a submodel $V[H], V \subset V[H] \subset V[G],$ such that $V[H]$ is a cardinal preserving extension of $V$, but  $C \in V[H].$ Now if $\alpha$ is a singular limit point of $C$ in $V[G]$, then it is the same in $V[H]$, and passing from $V[H]$ to $V[G],$ the forcing collapses  the singular cardinal $\alpha^{+\alpha}$ into the singular $\alpha,$ but it preserves $\alpha, \alpha^{+\alpha+1}$.
A: For question 2, some large cardinals imply the existence of such a forcing.  Suppose $\kappa$ is 2-huge, with $j : V \to M$ an elementary embedding, $\lambda = j(\kappa)$, $\theta = j(\lambda)$, and $M^\theta \subseteq M$.  Then using the embedding, we see $S = \{ x \subseteq \lambda^{+\omega+1} : (\forall \alpha \leq \omega+1) \mathrm{ot}(x \cap \lambda^{+\alpha}) = \kappa^{+\alpha} \}$ is stationary.  A standard argument gives that the following set is also stationary:  $T = \{ x \subseteq \lambda^{+\omega+1} : \mathrm{ot}(x \cap \lambda^{+\omega+1}) = \kappa^{+\omega+1}$ and $\kappa^{+\omega} \subseteq x \}$.  Also, there is a Woodin cardinal $\delta > \lambda$.  Forcing with the stationary tower up to $\delta$ below $T$ will produce a generic embedding $i : V \to N \subseteq V[G]$ with critical point  $\kappa^{+\omega+1}$ and $i(\kappa^{+\omega+1}) = \lambda^{+\omega+1}$, and $N^\delta \subseteq N$.  This has the desired collapsing effects.
