Connect Arduino to ESC with BEC (without using its power)
You state that you like battery life.  The Arduino will take a miniscule amount of power compared to the motor on the car.  It will probably take less current then even running the steering servo.  There are reasons for not using  the BEC power, but those are usually related to noise from that power.

Regardless, if you want to power the Arduino from a separate power source you need to make sure to use close voltage levels.  (RC can usually tolerate 4.8-6.0V or what was 4-5 NiCd/NiMh before LiPo became more popular.)

Your servo connection will just be Vcc, Ground, and signal.  The signal is the only thing connected to an Arduino pin, so we don't have to worry about how much power pins can provide.  The Vcc and ground will provide the actual energy to turn the steering servo, which can be quite high depending on the performance of the servo used.

For connecting with the ESC, you need to tie your Arduino ground to the ESC ground through the BEC circuit.  You will leave the power coming from the BEC open.  Then you provide the signal to control it.  By tying the grounds together, they are now at the same potential.  Even though you are using different power sources, you are providing pulses that will be understood by the ESC, because they are both referenced from the same point (ground.)