Not counting a few of the earliest machines that had the control unit mounted on the back door, it's common to find a few other things on the back door besides the mixer/spotting unit, the trip bank, stepper units and relays.
The large win replay counter is for wins that aren't associated with an in-line color, like a corners win or 2/3-in-blue. Because the unit is counting one or two fixed amounts, it's implemented as a many-toothed ratchet and cam with a couple switches running along the cam edge. When a switch falls in a notch, the payout stops.
The many-toothed ratchet/cam have a metal-sleeved center hole and it's mounted on a metal shaft. Lubrication should be applied and has often hardened - sometimes to the point where it takes penetrating fluid and time to get the ratchet/cam off the shaft.
The perimeter switches are adjustable, and ideally you don't move them otherwise you'll be in for some tedious counting or trial and error to get the payout right. The best approach is to get the ratchet/cam to rotate, then remove the c-clip and hold the switches and ratchet tabs away while pulling off the ratchet/cam assembly. Clean the hole and the shaft, lubricate and reinstall.
You don't have to worry about getting the unit to the home position, it will do that itself. Just make sure the unit steps around easily when the relay plate is activated.
On Magic Ring, this unit is mounted on the metal shelf inside the head. That may be the only game that put it there.
Usually the same style as the Jones plugs between the head and playfield/cabinet. Clean the plug pins and if you want the sockets (they are more accessible than the ones in the head). The game is most liberal when the plugs are inserted so the most wires are connected between the plug and socket.
Essentially a coil/plunger that's not connected to anything. It just makes noise. Early games tended to use it when the score unit(s) stepped up - apparently the noise from the score unit wasn't loud enough already.
The common failure is the plunger getting mushroomed and jamming in the coil. Remove the plunger, file off the ridge, and you're done.
Some of the late model games, especially the 6-card machines with a double-or-nothing feature, used a capacitor(s) and diode to implement a time delay circuit. The diodes don't fail often, but electrolytic capacitors can dry out. When replacing them pay attention to the +/- markings. If you put them in backwards they tend to fail quickly and spectacularly.
Likewise if you replace the diode, pay attention to which end has the band/ring on it. If you put the diode in backwards, on some games you've effectively swapped the +/- ends on the capacitor and the cap will fail. On other games a backwards diode will short the power and blow a fuse.
