That "silver box" is the mains filter and is connected DIRECTLY to the mains!!! The red part soldered to it is a MOV and needs replacing.

Lucky you are still here to tell the tale!

UNPLUG the machine from the wall outlet before doing ANY soldering work like this. If you are not confident working with mains power seek professional repair help.

To fix the flippers, watch some Youtube videos on how to use a multimeter AND even more important, how to interpret what that meter tells you.

The line filter and red disc Varistor should have plastic tubing covering the exposed wires. The varistor should be bright red and goes across the leads. It looks burned up. You should replace the varistor and use head shrink over the wires to prevent future shocks.

Look under the relay on the Solenoid board for cold solder connections. That is a common point of fail for flipper issues when both flippers are dead.

In solenoid test, listen for the flipper relay to activate. Could be transistor on solenoid driver board, cold solder or connector. I have had a lot of trouble with the harness connector contacts. I replaced every contact at MPU and solenoid boards with new and now machine is much more reliable. Try spraying Deoxit contact cleaner inside the connectors to see if gets better.
The varistor is ugly but not the issue.
The sound board issue is often failed IC chip. Need to verify voltages and listen for buzz while touching components. If replacing capacitors does not fix, make new post for more sound board ideas.

flipper relay is located on the driver board - square cube, it either works or doesn't and you should hear an audible click from it during power up

power it up with backbox open and listen for it to energize (click) as long as your mpu is working properly it should energize after/during boot up

flipper playfield connector is to left of the cube, maybe reset it and see if anything changes

have seen the traces from the relay to the pins disintegrate on several boards

if no click relay could be bad (usually is) would need to verify correct voltage is present at relay