Sell it to me and that will solve your problems

Seriously, looknat one problem at a time. Wish I was closer and could give you a hand.

For the player 1 issue, check the player unit. Is it a contact issue or a mechanical issue of not physically moving back to player 1 position.

For the gator issue, are the leaf switches adjusted so it senses what is happening properly? I dont have a schematic in front of me to see the details.

The 2nd Ball relay should fire when the ball drains and stay active while the bonus counts down to zero.
Nip It bonus count (resized).jpg
The Bonus Score relay should fire when the 2nd Ball relay does (for bonuses of up to 5000 points) and stay active until the bonus counts down to zero too. If it doesn't, check the switches in the path shown in red on the right side of the schematic.

Since the Player unit steps up but doesn't reset that could be the Make/Break switch in red:
Nip It Player Reset (resized).jpg
Either the Player Reset relay isn't firing or that switch isn't flipping to the other side.

Continuity isn't a reliable test for switches. It just tells you that the meter sees a path between the two probes, not that it sees a path through the switch you're interested in. In this case if you look at the schematic above, your meter is probably telling you that there is a path through the Player Up Unit step up and reset solenoids. If you want to know if the switch is opening and closing you need to check the resistance across the switch, not the continuity.

/Mark