[cheesy]

Welcome to the world of troubleshooting.
Possibly a dead cell in the battery but more likely a short in the starting circuit or the starter itself. I'm leaning towards the starter.

10 volts is a dead battery. You should see above 12.7, preferably 13.1. Clicking/buzzing is the starter solenoid when the voltage from the battery is too low to activate the starter. Wires getting hot is indicative of a heavy load on the circuit, possibly a locked up or shorted starter.

Try charging the battery and check it with a VOM. If you don't get at least 13 volts, try a different battery. With a fully charged battery installed, disconnect the big wire from the starter but don't let it touch any metal. Turn on the key and hit the starter button. Nothing should happen, no clicking/buzzing, no hot wires. If you get noise and sparks, you have a direct short to ground. You'll need to hunt that down and do some rewiring.

If there is no drama, turn off the key, reconnect the big wire to the starter, and repeat with the key on and hit the starter button momentarily. If you get the clicking/buzzing, stop and check if the big wire to the starter is hot. Also check the starter housing. If that's hot, too, that pretty much tells you the starter is toast.