Not sure why you have a capacitor on the flipper circuit. I'd remove it, but don't think that's your problem. You have power to the solenoid?

Capacitors can fail in a variety of ways, but the only way I can see that would cause the switch to just not work at all is for the capacitor to short. You can test that very easily while it's still soldered in. There shouldn't be continuity between the legs. If your multimeter has a capacitance setting, you can also test that; it may be less accurate while the capacitor is soldered in, but you should get something reasonably close I think.

The main discoloration I see in your photo is on one leg, and that looks like it's just flux that dripped down from the solder joint.

I can't see anything readable in the photo that would tell what capacitance that component is. It's obviously rated for 200 V, but if there were standard markings on it, they've worn off. Maybe there was a different marking scheme used when it was made and the "T 1" or "1 Z" are meaningful to someone with better knowledge. Of course, you could just pull the capacitor from the left flipper and measure it.

Note that if the capacitor shorted, it might have (probably) caused the flipper to get stuck, then eventually burn something else out. So you'll want to trace back and check the other components involved, such as the driver transistor, and resistors or diodes, if any (if you can't tell, I don't actually know the boards in this game specifically, so I can only write in general terms).

I don't know. Depending on the overcurrent condition, you might toast a transistor before the fuse blows. Besides, a shorted capacitor isn't going to put the load over the limit for the fuse, since it's just like closing the switch. So maybe the transistor or other component would still have a chance to overheat.

I'm not an expert in this area. I just know the circuits aren't all that complicated, so I usually just start at the point of failure, and work my way back until I find something that's amiss. You can start with the capacitor. You could even just take it out of the circuit; as noted above, the flipper should still work fine without it, you'll just get a stronger arc on the switches points.

Without the capacitor in the circuit, that frees you up to look at other stuff without worrying that the capacitor is interfering with your testing.