Where on an imaginary gravitational field line between Earth and moon, a mass would have no weight neither due Earth nor due to moon?
If you mean when the gravitational force is zero, then you just need to equate
FEarth and FMoon for a mass m and solve for the distance.
No, that 1/6 figure is only true about the surface gravity of the Earth vs. the surface gravity of the moon. It is not applicable everywhere in space.
I think you really do need to set up an equation using the equation for gravitational force. At least, I can't imagine solving it any other way.