DI or not (Was The New World not photochemical?), I would suggest this rule of preserving latitude extended to the grade - ie. their goal wasn't freedom to push the film around heavily in post but, rather, to capture and present images with the greatest amount of detail and the greatest range of contrast. Also, whilst there's nothing unusual about wanting a well exposed negative, to achieve this using only available light is most certainly unusual (and no small feat!)
This is most certainly true. The intention was not necessarily to give them leeway in post, but to preserve as much detail in the highlights and shadows as possible in the final image.
Re: the Z-Axis, this means rolling the camera, the visual equivalent of tilting your head to the side. Not a common tripod feature, it can be achieved on a steadicam, or handheld of course, but is most commonly found and used on remote head cranes. The most obvious example I can think of is the final shot of Irreversible (and Noe's work in general). Whether any use of the Z-Axis actually made it into The Tree of Life I can't recall.
But I don't think you've got this right. I've never heard of this type of camera movement ever being associated with the term "z-axis." In most applications, including cinematography and 3D animation, the z-axis refers to depth, and movement along the z-axis refers to moving the camera itself forward and backward (as with a dolly or a person). See here
(in the definition of "depth of field"), here
, and here
. Matrixschmatrix's explanation is correct.