Can somebody please tell me what the AE shift is for, exactly?

Auto exposure Shift (AES) does just that - it means you can leave the camera in the auto exposure mode but 'bias' it either one way or the other. So you can make everything darker or lighter. If you were filming in very snowey mountains for instance the camera would tend to underexpose and make the snow look grey. By shifting the exposeure a notch or two you can make it all look brighter.

tom.

Why wouldn't I just use use manual iris? Is the AE Shift there so I can keep the camera in auto mode for fine adjustment? And what does the AE shift actually control: the iris or the shutter speed?

You've nailed it - use manual exposure and forget all about the silly nancy-boy AE shift. It's really put there because it's so easy to engineer, and for some people who habitually under-expose (by having too much sky in all their shots, say) it's a fine customer tweak that they can set on day one and never touch again.

AE shift only alters the aperture, but note that when you reach f11 and ignore the ND silent scream, the camera will up the shutter speed to avoid over-exposure. In the AE shift mode this will still happen of course.

Leavi it alone is my advice. Lock the shutter (turn of auto shutter in the menu as well) and lock the aperture for every shot.

tom.