For your game start issue, check to make sure a switch is not open on your Delay relay. All coin and credit stuff goes through that switch.

For your light issue, check for open switch on Game Over Relay.

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Assuming you have credits on the credit wheel and you are trying to start it using the replay button, check switches along this path in red. When all switches are closed and you press the replay button the credit relay should fire, energizing the coin relay which energizes the reset relay through the game over relay.

If your credit relay is already firing then check the above mentioned relays.

Easiest way to troubleshoot switches is by jumpering each switch in question.

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Howard's right I didn't notice those switches to the left, I cut them off when I copied the schematic. I would check those coin switches first. More likely to be the problem than the others.