I'll answer the second question first.
It looks like it's previously kamikazed itself and then replaced.
Quoted from TicTacSeth:Hey ajfclark, I just pulled off the board to replace Q42, when I saw this up on Q77. Can’t be good. What does this control?
If you look at the second page of the manual, there's a list of solenoids and which transistors are their drivers:
Space Station Solenoid Table (resized).png
Q77 is a special solenoid #5, the left jet bumper. Special Solenoids are a bit special, in both complementary and derogatory senses of the word. They can be controlled by the CPU, and they are also controlled by a direct switch that comes in through 1J18. Because there's a single fuse for multiple coils that might fire almost at the same time (eg all the jets) if a single switch gets stuck on, the coil, driver, etc are all candidates to burn. It's enough of a problem that people sometimes add individual fuses for the special solenoids and this guy made a nice board to hold them that I keep meaning to buy: http://nvram.weebly.com/repair--conversion-kits.html
Why was it designed that way? I guess the original intent was that things that needed super snappy response would be Specials. As there's a direct switch, the coil is activated without waiting for the CPU to see the switch is close and fire the coil. Later, they realised the delay of a switch hit to the CPU triggering the coil wasn't long enough to outweigh the problems and through the System 11B machines the use of 1J18 was phased out. Special solenoids still exist in System 11C boards, but the switches to 1J18 are never connected. If you hold down a switch on a System 11C special, the coil fires once and then the CPU ignores it.
My #5 was cooked when I got my #space station so I wonder if it's a common fault with the way the playfield is setup - perhaps that post rubber that gets chewed out eventually holds the switch down.