Q1: In the past there was either no way to change the 1k x 1k default eye texture resolution (other than editing the OVR Unity plugin source), or you could configure it via OVRManager. But I think now this is done via render scale settings in OVRManager in Unity 5.4+. I do believe you're right that some of the official Oculus apps have carefully dialed in the use of higher resolution eye textures.
Q2: One idea is to remove the mipmaps from your 360 photo, so that it has no choice but to sample the full resolution. And there are other ways of introducing a mipmap level bias, but I forget where.
Q3: If you are using bilinear filtering, seeing the the edge between mipmaps can be very obvious as you move your head around. This is because your 360 photo is a sphere and the far side of your render frustum's depth buffer is flat. The answer here is either force a single mipmap level (or no mipmaps), or trilinear filtering which comes with a performance cost but is often worth it.
