Drop targets - check their switches or wipers, and from them to the board in the backbox for a broken wire.

LTG : )

When you say you hear all the solenoids popping loudly every time a flipper is used - do you mean multiple coils fire at once, or just that flipper sound itself is loud?

OK, I've never tried it, but here are a couple of random ideas - you could glue a piece of rubber to the coil stop that would help deaden the sound. Obviously would need to be replaced now and then. The other thing that might work is putting some light foam into the coil sleeve. Both of these would have an effect on performance of the game (the first would change how much the flipper can travel, and the second might affect speed of the plunger), but might help dramatically with the sound. Lots of times games will use rubber stoppers to keep various metal parts from smacking, which I think is usually to help with spreading the force of the impact, but it does help with sound too. Just a thought. If you try it, let us know how it works out. I don't think putting a pillow inside the machine is a great idea, tbh. Another option is to maybe line the cabinet with Mass Loaded Vinyl. Again, never tried it in a pinball machine, but it works for walls.

Google is your friend:

https://www.google.com/search?q=pinball+deadening+sound+from+flippers&rlz=1C1EJFA_enUS778US778&oq=pinball+deadening+sound+from+flippers&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIHCAEQIRigATIHCAIQIRifBTIHCAMQIRifBTIHCAQQIRifBTIHCAUQIRifBTIHCAYQIRifBTIHCAcQIRifBTIHCAgQIRifBTIHCAkQIRifBdIBCDg3MjBqMGo5qAIAsAIB&sourceid=chrome&ie=UTF-8

A lot of people have looked at it... most people have suggested various sound deadening foam solutions.

I wouldn't lay the foam directly on the transformer inside your game, and I wouldn't fill the whole game up to the playfield with anything, but I'd suspect you've got several inches for most of the pinball's internal space that you could safely put some sort of sound dampening stuff and not affect your pinballs operation.