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This is actually quite spot on, think of that dome as the concave mirror wrapping around you, and the top of the astroplate (that line with some blue around it) is reflected off of the dome around, and it even has lines to illustrate it. I don't think this picture even had this purpose (looks more like its modeling the december solstice), but it works as an illustration.
I don't think it'd work as you think. If I get around it tomorrow I'll make you a 3d view (night time here...) but it's the same reason why old videogames had a separated "sky box" for the far scenery. Perspective messes things up. None of the options you cite would really work once you're a bit sideways. You'd see either far stars getting too close to each other, or close constellations deformed.
I don't quite know what you mean here, but I am assuming you are referring to the angle of view problem with stars and the moon that I just made a thread on (asking about how it works): https://forum.tfes.org/index.php?topic=6821.0
Otherwise, I'd have to see what you mean.
So, here are a few pictures from the model I've done.
I had to eyeball the proportions from your pictures, because no dimensions are given (figures...)
In the side view you can see the earth and the astroplane.
The darker area in the astroplan is a rough approximation of the visible sky from a point on earth ca. halfway "south" of the Astroplane center. From your map it would be like Florida? Egypt? beats me.
The other picture is a first person view from that point at ground level. As you can see, perspective has a massive effect.
I lack the time now to render the concave / convex alternatives, but, trust me, you'd get similar problems.
Crappy quality, but the attachment size costraints are crippling