Yes, on the controller large Red and Black go to the battery. the battery in this case would generally be a RC hobby sort of battery, the kind you see most often these days are LiPo batteries, they are listed as 1S, 2S, etc. Each S is 3.7V, so a 4S Lipo is a 14.8 Volt battery. The size of your motor is going to determine what rating of battery you will need, as they come in various flavors of milliamp-hour, C (maximum safe continuous discharge rate), and S (number of cells which also determines voltage)
And also correct on the servo connector, it has output red ~5VDC + Black ground, and you can think of white as control or data or whatever you would like, this would be the PWM output from the arduino.
As far as motor hookup don't really worry about it too much, three wires, hook them all up, it will work, just jitter, or run backwards, but it will not damage the motor or the controller in any way if hooked up incorrectly.
Generally when you see people trying to interface a servo with an Arduino they are trying to power the servo off of the same source as the Arduino and they run into issues with electrical noise, that is why you often see discussion of using a separate power source for each. If you use the red/black on the servo connector from the ESC to power your Arduino I believe that everything will work just fine (I have personally done this with no issues at all, the ESC BEC provided a very clean power source and created no issues with noise for me at least). In this case there is no servo to create noise and the Arduino is isolated from the motor by the ESC, but as far as programming is concerned just think of the ESC as a servo.
One question I have about your ESC, I know you say you really have no documentation, but was this ESC advertised as an ESC for a car/truck, or for a Plane/Heli, or was it not specified?
The reason I ask is this, like was discussed earlier most every ESC requires being held in a certain position to arm. If it is a car/truck ESC they generally have reverse as well as forward (and brakes too), so it needs to be held at 1.5ms to arm (90 degree position if you are thinking about it from the servo library point of view). After armed 1.5ms thru 2ms (90-180 degrees) is forward speed, dropping below 1.5ms is brakes (0 thru 89 degrees or so), returning to neutral (1.5 ms or 90 degrees) from brakes and then dropping below 1.5ms again is reverse.
For a plane/heli ESC you will need to hold 1ms (0 degrees) to arm and it is all increasing motor speed from there.