Are you taking about the orange peel, the orange juicy part or the whole fruit?

If you consider only the "orange peel", if the center is 0 and r=1, then the 1D version is just two points {-1} and {1}, not only one point.

If you consider only the "whole orange" then the 1D case is the segment [-1, 1], if the center is 0 and r=1

In both case it is equal to the 1D "square" equivalent.

Really depends on who you are asking for a definition. E.g. there is the term "point sphere" for the special case of a sphere that is a point.

Is there any interesting result that comes from casting a point as being any shape (presumably the point-sphere is also a point-dodecahedron, etc.)?

Define a topology on any set you'd like and a sphere is just the set of limit points of any open ball.

The 1D case is a line segment, not a point. A point is the 0D case and indeed a degenerate case.