I ran into something kind of similar with a Gorgar. Game was on free play, when you started a game, it would add another player (I think that's what it was doing, my memory is hazy). When you hit one of the drop targets, it also triggered a rollover, and there was at least one other strange switch thing going on. It ended up being an issue with the wiring on a drop target bank. I found it by comparing the wiring on the two banks and noticed that one of the diodes was attached to a different screw than the others. I'm still not sure exactly what was wrong, all I remember is changing one thing on how the bank was configured and everything worked correctly.

I see on Tri Zone that Switch 35, "Zone drop target series" is in the same row as the credit button. I think this is for sensing when all four targets are down. Check the wiring and diode for this series. It is spread across the four drop targets.

Also look at all the switches that are in Column 1 with the credit button. Check the diodes, check nothing is stuck closed. Pay attention to the tilt and how it is wired, sometimes people mess that up. Same with the ball roll tilt, make sure the leaf switch isn't bent closed to the frame. Check the slam tilt on the door to make sure nothing is shorted or corroded. Finally, make sure the coin switches are not messed up or wired wrong. Often they get bunged up and mangled from people adding credits.

Unfortunately the fact that you can't start a game makes it harder to track this down as you can't see any weird switch triggering. When in test mode, pay close attention to what happens when you trigger switches. Make sure all the drop targets are up, then start triggering switches. Watch to see if one switch triggers followed by another very quickly.

Finally, I have a Tri Zone. If you need any wiring double checked let me know and I'll take a look.