Greg,  Although I don't own a "Musical Casket" like yours, I can still
make some educated guesses about your instrument:

The "large rectangular opening" you mention is where a manually
operated swell shutter would have originally gone.  The wood blocks
are part of its hinge mechanism, possibly spring loaded I'd guess.
It would have been opened to emit more sound (volume) or closed for
quieter passages.  It would have no effect on the actual "playing"
of the instrument, so is probably not your priority for repair.

The white rubber drive wheels were shifted by a lever (I think) in
order to engage the rewind spool.  Assuming the drive can be placed in
its normal playing position, again, these would not affect "playing".

I would be suspect of the bellows cloth, even though you observe that
the "bellows material is okay".  It's likely that there are a multitude
of small leaks, especially anywhere the fabric folds.  But if there are
no obvious leaks, it would usually make some sound upon a fast
cranking.

The bellows arrangement is usually like this: a couple of exhauster
bellows move the air from inside the reservoir bellows.  This is
accomplished by leather flap valves on both the intake and exhaust
ports.  (The flap valves between the exhauster and reservoir may be
inside and not readily visible.)  The reservoir collapses to hold and
even out the vacuum until a valve opens (the slots in the paper, in
this case) and the air rushes in through its reed, making the sound.
The reservoir has some device to prevent it from collapsing too far:
a relief valve of some sort, probably what you describe as a "wooden
gate".  The relief is closed until the reservoir "nearly" collapses,
then opens and admits air.

To test the air system, I find it easiest to place some removable tape
across all the valves except one or two.  That way you don't need a
roll installed, and can concentrate on the bellows and check valves.
Crank and make sure that the exhauster bellows are moving and air is
exiting their flap valves.  Then check to see that the reservoir is
collapsing and/or the notes are playing.  If not, find where the air
is entering.

Have fun!
Todd Augsburger - Roller Organs
http://www.rollerorgans.com/