Start fixing switches. The optos may need cleaning, or more work. They can be checked in switch edge test.
Did you watch if the ball launch kicker that sends the ball up the shooter lane to see if it strikes the lane switch ? If the lane switch is closed the game thinks a ball is there and kicks again. And be sure none of the ball trough switches are sticking. Go into switch edge test and roll a ball over them and see if they open and close.
And since you just got the game, you might run over all the switches to check for switch matrix issues. Any switch wired wrong, diode/wires/switch lugs shorted together, or shorts to switch wiring. To do a quick check, Williams uses a basic coloring scheme in their switch matrix. White and green. For a quick look the color of stripe won't matter. On a micro mini switch with three lugs, two close together and one farther away, the two close together the outside one gets the white wire and black end of diode, middle one gets the green wire, and far one gets the silver banded end of diode.
The coil not working - check the coil for broken wires, especially the tiny ones coming off the coil to the lugs, check too for poor solder joint. Then check continuity of wiring from coil back to the power driver board.
For what you have going on, no need to worry about fuses.
LTG