A Mega can be powered with 12volt on the DC socket (safer) or V-in,
but at that voltage you can't power much else from the mega.
I don't know what "a host of LEDs" means, but about 150mA is the limit.
A 5volt buck converter connected to the 5volt pin (and maybe the LEDs) could be the answer.
Post a diagram, so we can verify.
The sensor is easy.
White to a digital pin, with pull up enabled in pinMode.
Black to Mega ground.
pinMode(sensorPin, INPUT_PULLUP); // LOW = active sensor
Use twisted pair for that 6Ft of wire,
and an optional 100n ceramic cap from pin to ground if in a noisy environment.
Leo..