Well, howdy! Welcome to the party!
The only difference with the '13 version versus the '12 is the electronic steering damper. Obviously that won't affect anything with the motor.
Your coolant reading is likely due to you not properly connecting the coolant sensor or it's damaged on the replacement motor. The fact that it's showing anything on the cluster at all won't tell me the wiring is fine and the sensor itself is faulty. That's the first thing to look into. Swap it out from the other motor and see what that does.
As the the starting issue, sounds like a bad connection at the starter relay. The clck is the relay engaging but it grounds out and disconnects the power source so when the button is released it restarts the power-up sequence of everything. The fuel pump primes for 4 seconds every time the power is initially applied to the ECU. So, the it's losing power while the button is pressed. Make sure all the harness ocnnections are proper and look at the starter relay connections as well as the starter motor itself. The starter on the new motor may be old and damaged and what's causing the grounding issue. I'd receommend swapping the starter from the other motor as well.
