If the sling is firing when you get into solenoid test mode, there's a couple possibilities, IC 5 likely wouldn't be it. This one is going active when the solenoid activation signal from IC5 turns on, so the PIA is pretty much working fine. This also rules out Q12, as a stuck transistor wouldn't care which mode the game was in, it would fire as soon as you powered up the machine.
#1 possibility is just a switch adjustment at the sling or something goofy in the wiring. Easy enough to check. On the left side of your driver board, pull the connector closest to the top. This will disconnect the special solenoid switches. If the pop bumper de-powers, it's wiring. If it stays active, it's driver board.
#2 is the driver board. Could be either IC7 or IC9, with an off chance of being R6 or C6 but I rarely see them fail. In audit mode, I'd check the logic states at IC7 pins 8,9,10 and IC9 pins 11,12,13
#3 I's also very unlikely, but ST6 that fires that particular special solenoid actually comes across from the CPU board PIA chip IC18 through pin26 of the interboard connector. Check the logic state at that pin too.
I have too much of a headache right now to dig up all the 74xx logic states to figure it all out, but those are your primary suspects in this situation.
-Hans