Yes, the 3-1 has a BEC and steps the voltage down to 5 or so volts for the rx and servos.

Hopefully, you removed the power wire in the tail ESC per the instructions so the 3-1 wasn't fighting the tail esc. Most ESC's have a BEC.

Yes, the c-12 should be enough. Some guys have measured the amperage to the main motor and its never above 10 amps, more like 6. If I were you, I'd buy a c-7 and use it for the tail. (cheaper) and move the c-12 to the main motor. Just remember that only one ESC should power the rx, and I wouldn't try to power it through the gyro. (disable the tail's BEC per instructions)

Do you have a way to connect a 4.8 volt rx batt to check the receiver without the 3-1? (Just run the servos) Maybe its still good and only the 3-1 is fried.

Just re-read and you stated smoke from the rx. You're just gonna have to experiment with replacement rx. Good Luck...