The thousands lite is controlled by a single normally open switch on the HB relay (or thousands relay), which is located in the control bank and may be hard to see. That switch may be gapped too closely and making contact when it shouldn't??? The HB relay only has 3 switches. Look for one with black/yellow and white/black wires attached. The circuit can be seen at location 6B to 6C on your schematic. Also make sure the control bank carriage is well seated and the wing nuts are secure.

If that fails, it might be possible that the 9-position switch on your hundreds reel is gapped too closely and vibration from the score reel is causing the switch to momentarily conduct and the HB relay to pull in prematurely. That circuit can be found at location 3E to 3H on the schematic.

On the kickout hole issue, might want to inspect to make sure the Kings hole kicker arm is not lightly binding on something and causing the kicker arm to hang up and stay up longer than the other kicker arms.

Good ideas. IME, all kinds of wonky switch behavior can occur on a Gottlieb when a switch bank carriage loosens up and one end slips slightly out of place.

If the other kicker arms are functioning normally, it does sound like the Kings eject arm may be hanging up.