$\newcommand\possible{\diamondsuit}\newcommand\necessary{\square}$The
forcing modalities defined for a given definable class $\Gamma$ of
forcing notions are

- $\varphi$ is $\Gamma$-possible, written $\possible\varphi$, if $\varphi$
holds in
*some* forcing extension by forcing in $\Gamma$.
- $\varphi$ is
$\Gamma$-necessary, written $\necessary\varphi$, if $\varphi$ holds in
*all* forcing
extensions by a forcing notion in $\Gamma$.

The main question is, for a given class $\Gamma$ of forcing
notions, what are the modal validities of $\Gamma$ forcing? For
example, which modal assertions are correct for c.c.c.-forcing?
for collapse forcing? For proper forcing? And so on. To give an easy example that may illustrate the point: when $\Gamma$ is closed under finite iterations, then $\necessary\varphi\to \necessary\necessary\varphi$ is valid for $\Gamma$-forcing.

I introduced these forcing modalities in A simple
maximality principle, because the maximality principle itself is easily expressed in
this modal language as the assertion that every possibly necessary
statement is true. Benedikt
Löwe and I proved in The modal logic of forcing that the ZFC-provably
valid principles of forcing, for the case where $\Gamma$ has all
forcing notions, is precisely the modal logic S4.2. That work made
use of the concepts of buttons and switches:

- $\varphi$ is a
*switch*, if $\varphi$ is necessarily possible
and also $\neg\varphi$ is necessarily possible. In other words,
you can turn $\varphi$ on and off by further $\Gamma$ forcing. $$\necessary\possible\varphi\qquad\qquad\text{ and }\qquad\qquad\necessary\possible\neg\varphi$$
- $\varphi$ is a
*button*, if $\varphi$ is necessary possibly necessary. In
other words, you can force $\varphi$ in such a way that it remains
true after any further forcing. $$\necessary\possible\necessary\varphi$$

Our more recent work is Structural connections between a forcing class and its modal logic, joint with George Leibman, expands
on the general connection between the structure of the forcing
notions in a definable class $\Gamma$ of forcing notions and the
modal logic of forcing to which it gives rise. In order to do
this, we introduce a number of other types of modular control
statements, including ratchets, weak buttons and others.

For the case of collapse forcing, we establish in the paper the
following (thm 24), and this answers your question 1:

**Theorem.** (Hamkins, Leibman, Löwe) If ZFC is consistent, then the ZFC provably valid
modal principles of collapse forcing are precisely the assertions
of the modal logic S4.3.

This doesn't mean, however, that every model of ZFC has S4.3 as
its validities. For an upper bound on most of the modal logics, we
had proved in the paper that if a class $\Gamma$ of forcing
notions necessarily has a family of independent switches, then the
modal logic of $\Gamma$-forcing is contained in the modal logic
S5. And what Benedikt and I had noticed earlier was that for
almost all the natural definable classes of forcing $\Gamma$, we
were able to identify families of independent switches. For
example, with c.c.c. forcing, we can always make the continuum
equal to $\aleph_\alpha$ for very specific $\alpha$, either even
or odd (or to have some particular binary bit pattern), and so for
any class $\Gamma$ containing the forcing to do this, there will
be independent switches. In particular, there will always be
forcing extensions $V\subset V[G]$ having different theories. And
we had similar switches for most of the standard classes of
forcing.

But in the case where $\Gamma$ consists of all
collapse-to-$\omega$ forcing, we couldn't seem to identify any
natural switches. Thus, we wondered whether there necessarily were
switches. An extreme negative case of this would be a forcing
extension $V\subset V[G]$ where $V\equiv V[G]$ are elementary
equivalent. And so we had asked whether this was possible.

At the Będlewo set theory meeting in 2007, I mentioned the problem to Paul Larson, who proposed an approach by which we should add a club to a suitable large cardinal and then collapse cardinals between members of the club. At Będlewo, we mentioned the idea to Bill Mitchell, who later carried out such an idea with his proposed solution, On a
question of Hamkins and Löwe, but afterwards backed off of the proposal. (Meanwhile, see Mohammad Golshani's answer for later news on this approach.)

So to summarize the answers to the four questions:

Question 1 is answered in full by the theorem stated above. The provable validities of collapse forcing are precisely S4.3, and this is different from provability logic.

For question 2, if you mean consequences of the situation where necessarily $V\equiv V[G]$ in every collapse extension, then the modal logic trivializes, since if $\possible\varphi\iff\necessary\varphi\iff\varphi$, then all the modal logic in effect evaporates.

For question 3, I'm not sure about the nature of the model apart from the lack of switches, which I find fairly strange. It is a very different situation than occurs with most other classes of forcing, since with other kinds of forcing we can generally affect the theory of the resulting model.

For question 4, we point out in the "Structural connections" paper that all the other natural classes of forcing do not have this situation. Indeed, it was our inability to provide switches in the class of collapse forcing that led to the question. For example, it is a good exercise in forcing to show that there are switches for c.c.c. forcing, or for countably closed forcing, etc. Give it a try!