for your pixelated HDRI, the problem is visible in your first picture with the texture properties. There is the original HDRI resolution, then the displayed and the max ingame res. The latter ones are the cause of your problem, and you manually need to raise the ingame resolution. For that, click that tiny arrow below the compression settings line, then it will expand and give you access to an maximum ingame resolution silder (default value = 0).
This one you need to raise to the closest number (it´s for thei height), which can create a 2:1 texture.
F.E. if you set it to 2048 (height), then this will create a 4096x2048 cubemap from your HDRI.
You cannot raise it higher than your actual imported texture, and unfortunately, you cannot use values between those steps.
For the displayed resolution, you would have to set it to 4096, resulting in a 8192x4096 texture, since the next higher would be 8192, and your texture is only 6107 high.
You other problem seems to come from your material being linked to the camera position. I have a skybox material, that does not have the camera position, therefore it is tied to the center of the sphere/object center. Looks like this:
Not sure, if that solves your problem, but with this variant, you should be able to adjust the height by raising or lowering the skybox mesh ^.^