I'd like to offer an alternate perspective to James L. Wolf's excellent post:

Don't change the pressure pads...use a machine that intentionally pushes
the pressure pads out of the way and develops its own tension. Such
machines aren't being made new any more, and I don't know the entire scope
of which machines did that. I know my Nakamichi Dragons do that. I think
that the Nakamichi MR-1 does that as well.

What I love about the Dragon is that it also does auto azimuth of the play
head. There is a model that Parker Dinkens uses that has manual play
azimuth adjustment--better in some instances, more labor.

While I can charge for remounting cassettes into new shells, it's really
nice to pop in a cassette and have it play well the first time.

I would definitely change out the shells if they are at all warped.
Corrosion on the screws doesn't bother me unless there is corrosion or
water/mud/slime/mold/mildew/carcasses inside the cassette.

If there is anything to either steadily or--even worse--intermittently
impede /disrupt  the motion of the tape that needs to be addressed. Azimuth
errors introduced by the shell are almost fully compensated for by the
Dragon's mechanism,  but not always--especially if there are cyclical
errors as opposed to constant errors.

If the cassettes got really wet, there might be residue on the slip sheets
that could induce stiction. That requires a new shell, in my opinion.

Also, beware of mold/mildew/fungus/carcasses for health reasons. I'm not an
expert on this, I've been told to be careful. Oh yes, animal feces
(including bird do do) can have nasty stuff in it.

I enjoyed your description of the storage conditions. I just saw on another
list someone mentioning this type of storage in regards to the library of a
very famous musician. GAK!

So, to answer your first question.

Examine, shake and manually spool some of the tape (pencil in hub). If no
contamination is visible in the window, nothing falls out, and the tape
moves smoothly without stiction and the cassette is not obviously warped,
play it in a machine that doesn't use the pressure pads, preferably with
one with adjustable play azimuth.

Also, if the tape smells bad, change the housing and carefully examine the
tape--but beware of possible health effects. I got sick for a week with
allergies once after visiting a very old, musty basement.

Sticky shed, while not unknown in cassettes, is not all that common. If the
layers don't freely unroll, further examination is needed.

I don't think putting the tape in a different style housing would have
much/any impact on the tape reproduction. HOWEVER, the azimuth might drift
between the two housings. Another reason for adjustable play azimuth.

The majority of the cassettes--and sadly a portion of the reels--I've
restored, especially in "oral history" collections were poorly to very
poorly recorded. I did 30+ cassettes for a small archive and the tapes were
recorded in many instances on very inexpensive portable recorders with
built-in mics. Air conditioning, dogs, people, and kids were all present
for some of the time, and made their presence known on the tapes. Screen
doors were heard to slam.