Is there any way to design a lens or system of lenses where I see the same part of the image below the lens, irrespective of direction from where I am viewing. In the image uploaded, there is a black dot near the focal length of the lens. This black dot will be visible magnified to me if I look at it from above. but if I look at it from sideways, the focal point will shift. So, is there any lens or lens system that can be designed where I can see the same magnified black dot, whether I am viewing the dot in direction perpendicular to it or at 90 degrees in either direction? A lens or lens system where focal point is not dependent on the direction of view? The light rays can be assumed to be collimated as viewing will be from a distance.
You're on the trail of a "fisheye" lens. If you spend enough money, you can get a lens with greater than $\pi$ Field of View in elevation. (Yes, for real) Here's one way to look at it: if you can see the same image on the focal plane wherever you are, then the focal plane can "see" you wherever you are.
Let's take a look. It is very useful that you've provided the images. Basically what you say that every point on the surface of this "lens" will "emit" in every direction. Well I guess one just need to glue a paper on this lens so that it will nicely scatter the light, i.e. emit in every direction.
Will the image look like a point or a circle from the distance? More like a circle of course. But this is a consequence of the finite size of the aperture of your eye and restrictions which you put on the "lens".
This is probably not the kind of device you wanted. But the one which fits your images