When a gravitational wave hits the earth, does the planet oscillate in place for the duration, or is our position in the cosmos displaced, or something else altogether?
The actual space in which the planet resides stretches and shrinks as the gravitational wave passes through it. The fabric of space itself is the medium that the wave travels through.
However, the affect is incredibly tiny, even though it was generated by two black holes colliding. The size of the distortion experienced here on Earth is 1000x smaller than the width of a proton! It's mind boggling.
I think I remember hearing that there is immense distortion in the area immediately around the collision, but I'm not certain.
The resolution of the detector is 1/1000th the width of a proton. The signal they see is much bigger: the length of 4km long arms stretch and then shrink about 1/10 the width of a proton. How do they know it is really space/time changing and not an earthquake? They have two detectors over 1000 miles apart from each other:
That is incorrect. What you describe is a dipolar oscillation. The simplest gravitational waves woudl have a quadrupole moment: stretch in one direction while contracting in the direction perpendicular to it. Have a look at https://en.wikipedia.org/wiki/Gravitational_wave where there is an animation illustrating the simplest case.
