If that's a close up of the bottom right board, that's the driver. Probably just heatsink paste or something, not actually corrosion, unless it dripped all the way from the MPU above it (top right). Battery corrosion probably wouldn't affect the displays anyway (could be just a bad connector).
Displays are controlled via the two right side connectors on the MPU. The interconnect cable between the mpu and the driver could cause the solenoid issue, it can be quite flaky. Reseat both with game off and see if anything changes.
Flippers (and probably other solenoids, but I don't have my schematics handy) are also controlled via the game over Q relay, could be the contacts need cleaning/adjusting.