Should I replace the nozzle with this one so that I know what nozzle I've got and so I'm sure this isn't a problem?
It appears the default nozzle size is 0.3 mm. The linked nozzle is a direct replacement.
How do I fix the extruder?
The "feed knurl", usually called the extruder gear is attached to the stepper shaft using a fine grub screw ("threaded pin" is a bad choice of words...). The black hole you mention is the hole of the screw (it should be in there, usually an allen key or flat screwdriver should be used).
Should I replace the filament with new 1.75 mm PLA? If so, why?
If the PLA has been open for a year and not stored properly it could have taken up moisture, I would buy a new spool (3.0 mm) of filament, not 1.75 mm.
Is the unconnected fan important and if so: what do I do with it? There's no remaining wire to connect it to.
The fan doesn't look as if this is a default fan, this fan is mounted onto cooling fins probably put there by the previous owner to cool the extruder stepper. Maybe the printer has been enclosed and used to print ABS at elevated temperatures. For Printing PLA cooling the extruder stepper would not be necessary as it doesn't require an elevated print temperature.
How tight should the 4 screws that hold the filament between the extruder and the rolling disk be?
The 4 screws need to be tight enough for the extruder gear to leave indentations on the filament, but not too deep.