the movement relative to the eye, before the modulo bit resets the positions of the vertices
The modulo should be the only thing moving the vertices. As far as movement relative to the eye, aside from the standard ModelViewProjection matrix work, nothing should be translating relative to the eye. The camera moves, and the vertices only change position when the modulus changes.
The grid cells need to be odd because each level of detail has twice the number of vertices as the next level, and they need to line up. Like this:
That inner level has 4 vertices along its edge. See how that hits the middle of a quad in the next LOD? By extending it to 5 vertices, the LOD edges always line up.
All grids are 2^n - 1. The reason for this number is that it's generally preferred for textures to have dimensions that are a power of two (2^n), but since odd numbers are required, the vertices only sample from 255 of the 256 pixels available.
Don't get discouraged!
Look at this video: http://www.youtube.com/watch?v=2xRuZ4byxVk
Pay attention to the outer levels as the camera moves. See how they only "move" when the camera has moved far enough for the modulus to index the next pixel? They don't slide around with every change in eye position.