If you want to use an ESC, yes you convert your code to using the Servo library. That's really easy though.
BEC is "battery eliminator circuit", i.e. it's a 5V power output from the ESC, intended to power the radio receiver and therefore eliminating the need for a separate receiver battery. You should be able to power your arduino from the BEC output but if the BEC is linear, it will be extremely inefficient.
If you don't need bi-directional control, you can just use a logic-level MOSFET and a catch-diode (motors are inductive!) and control the power with PWM to the fet gate. If you want bi-directional control, you typically make an H-bridge (4 FETs and a bit of level-translation circuitry) and control it with PWM. By the time you've done that though and made sure it's shoot-through proof (snubbers, etc), you would probably be better off just buying the ESC.