Yes, all the coils are connected together, that's how they get power.
They are individually grounded by a transistor on the driver board, that's how each one fires on its own.
A shorted diode on one coil will cause the others not to work if it blows the fuse. Games like Flash have a single 2.5A slow blow fuse on the power supply that controls all the playfield coils. If it is burned out, you'll get nothing.