1. 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?

That is not really necessary. The exact nozzle size does not really matter.

Should I replace the filament with new 1.75 mm PLA? If so, why?

No, there's no reason to do that. Going from 3mm to 1.75mm filament usually requires replacing the entire extruder.

How do I fix the extruder?

Tighten the grub screw (the "black hole") with a hex key.

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 appears to be cooling the stepper motor. This shouldn't really be necessary, the stepper should be fine without any additional cooling. People tend to "upgrade" their printers by adding unnecessary bells and whistles so it is possible the original owner added this fan just because they wanted to, and not because it is necessary.

How tight should the 4 screws that hold the filament between the extruder and the rolling disk be?

Tight enough that the extruder grips the filament and is able to extrude it consistently. If it is too tight it might deform the filament too much which could cause printing problems but it should be obvious if this is the case. There is a wide margin here so don't worry about this too much.

1. 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?

That is not really necessary. The exact nozzle size does not really matter.

Should I replace the filament with new 1.75 mm PLA? If so, why?

No, there's no reason to do that. Going from 3 mm to 1.75 mm filament usually requires replacing the entire extruder.

How do I fix the extruder?

Tighten the grub screw (the "black hole") with a hex key.

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 appears to be cooling the stepper motor. This shouldn't really be necessary, the stepper should be fine without any additional cooling. People tend to "upgrade" their printers by adding unnecessary bells and whistles so it is possible the original owner added this fan just because they wanted to, and not because it is necessary.

How tight should the 4 screws that hold the filament between the extruder and the rolling disk be?

Tight enough that the extruder grips the filament and is able to extrude it consistently. If it is too tight it might deform the filament too much which could cause printing problems but it should be obvious if this is the case. There is a wide margin here so don't worry about this too much.