Fuses good?

This era Williams notorious for poor drop targets, luckily there are modern day superior replacements, get with HHaase he can help you there.

For your kick out coil, perhaps diode was swapped or your transistor is bad.
http://www.pinwiki.com/wiki/index.php?title=Williams_System_3_-_7

^ link on good HOW TOs

Do you own a DMM ?

Replace the 40 pin interconnect if it hasn't been done.

Male and female side

I've redone a number of games from the 6&7 era. The main suspects on these old boards are the 40-pin interconnectors need replaced (mandatory), the ram/rom sockets need replaced, and any AMI brand chips/ICs need replaced, especially the large 40-pin PIA chips (6820 or 6821) need replaced. If you have multiple blown transistors it can be from the game locking up due to problems with those prior 3 items I listed. The blanking signal can't do its job and multiple coils can get locked on because of it and blow transistors. I know it sounds like a lot of surgery to do on the board but it really is the only way to feel confident about the issue going away for good. Its really only $20 worth of parts... more of a timesink than anything.

I cleaned my drop targets on my Flash as they would not reset once all knocked down. as you can find on the forums, the drop downs are all connected in series so once all knocked down, they complete the circuit to tell the electronics to fire the solenoid. The horseshoe contacts are pretty sensitive and a couple had to be readjusted after I cleaned them and then they worked.. However, once I put them all back together, I blew up the TIP for the center 3 targets (lots of smoke and the transistor package was blown apart!). Turned out I didn't solder the solenoid wire back to the solenoid - <duh> and the wire shorted to the bracket, so after replacing the TIP, the same thing happened again.... <ugh!> turns out I reversed the wires on the solenoid, so the diode would always conduct and I basically had shorted the 35V though the TIP... Long story short, if your coils are burned or the TIPs are getting blown, make sure the wires are not reversed on the solenoid. I haven't seen a diode failed shorted, but that could also be the case.

It's a great game once it gets working!