Tom,


My question a

1) Given the burnt and melted condition of the capacitor as shown in the
photos, is it possible (or likely) that the ceiling fan motor itself is
also bad?;


Possible yes, likely no.


2) What does the capacitor do;

Without going into theory these are "motor run" capacitors and maybe a
"motor start" capacitor. They help to shift the phase of the magnetic field.
You get more torque to spin the motor.

and do capacitors sometimes just burn and
melt like this one did without that being caused by something else such
as a bad motor?

Yes, capacitors fail with age. They are a common point of failure.

Dave M.