From what I have read, the signal line should only draw a few milliamps at most. However, as Arduino can safely sink or source up to 40mA on it's pins, if you want to limit the draw to say half that just use say a 270 ohm resistor in series with the signal pin to provide a current limit. On a 5V line no more than 18mA should be able to flow in or out per pin.
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.)
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Usually RC servos work on a pulse principle. The input idles low. A pulse (TTL-level) of width 1.5ms puts the servo at mid-position (using feedback from an internal pot), 1ms brings it to one extreme and 2ms to the other.
The pulses need to repeat at some frequency, 50Hz for example, the exact frequency is not too important. The same receiver can supply, in turn, pulses to several servos since the maximum pulse time of 2ms is much less than the 20ms period.