When you hit the flippers do the flippers flip?

Start with a switch test bang on the playfield and see what switched are closing, could just need adjustment.. not kicking out a ball they could be magnetized and sticking in the trough

Sounds like a few issues. First do a switch test and hit smack the playfield on different spots, use your phone on slow mo to record the reports on screen. The points with the flips sounds like a leaf switch activating with the vibration.

The balls not coming out could be a trough optó or it could also magnetized balls.

Is the burnt part a resistor on the sound board?

There are several people on here that can work on boards if you don’t feel comfortable with it. Chris Hibler is one of them. He has a website you can look at. It will need more than just the resistor though from the look of it. Might be cheaper to get a replacement

To me it looks like a flasher or coil got shorted to your switch matrix.

Inspect under your playfield and all coils and lamps and switch wires and housings. For broken or touching wires. Even before you get your board fixed.