The short answer is NO, you can't do that. Consider the size of the resistor needed to dissipate all that wattage and the fact that the current is not stable so it is practically impossible.
What you need is a DC to DC converter.
First figure out the min and max values that paddleX can have. I assume if the paddle is two dots wide then its min,max will be 0,6 Then figure out the min and max of your pot, typically 0,1023 for a a 10 bit adc. Now to map your pot value to paddleX value you'd use the command :
paddleX = map(analogRead(A0),0,1023,0,6);
here is the map documentation : ...
For your needs you don't need actual UARTs. You can do it all with normal GPIO pins and one (or more) interrupts.
UART communication is characterised by having a "start bit" at the beginning of all bytes. That is, in TTL terms, a transition from HIGH to LOW. You just need to look out for that transition.
Use Pin Change interrupts on 6 ...
Is it as simple as hooking up a resistor in series with the Arduino to bring the voltage down to about 7V for the Arduino?
No, it certainly is not.
Can I make an assumption for the current to solve for the resistance?
No, you cannot. You never assume anything in electronics without a very very good reason.
You require a suitably rated voltage regulator (a ...