Yeah, sounds like you just need to close the coin door... the goalie and soccer ball are motors, not solenoids. So, your solenoids are probably fine.. but that doesn't solve your start button issue.
You can probably trace the start button wires back to the boards and make the connection between the two pins to see if it starts a game right away... if so, that should tell you that you have a problem in the wiring. If it doesn't fix it, then it may be on the board. I don't have the papers handy to determine which pins are used for the start button.