If a point lacks any dimensions, it has dimension 0, which is very well-defined and consistent
This isn't an object with the dimensions of nothing this is an object with "no" dimensions.
Okay, what you're doing here is confusing the concept of "dimension" and "dimension", which, to be fair, is quite confusing. In a mathematical sense, "dimension" is a property of a subset of a linear or affine space, and it is normally a non-negative integer which is the minimal amount of axes of any coordinate system that can fully encompass all points in that set. So the dimension of any set S consisting only of a single point P is 0, since there is a coordinate system with 0 axes (with origin at P) that is enough to fully describe all points in S. Let's not go into Hausdorff dimensions for now.
If you were to write the dimensions of a 1 dimensional object it would either be a single number or a range of numbers.
I don't know what you mean. Example please?
You cannot write the dimensions of a 0 dimensional object because it has no dimensions. It exists in a non-dimensional space.
Sure you can, just as you can write the "dimensions" of a 1 dimensional object. And while the point itself is 0-dimensional, the space it is inside is not necessarily 0-dimensional too.
Its dimensions are not defined because you are essentially dividing by zero.
No, dividing by zero does not happen.
Cannot be moved because that would require it to have definition.
It has a perfectly valid definition. Let's use the point (2,3) in the 2-dimensional Euclidean coordinate space. You can translate that to (5,4) without any problems.
I cannot express this well but one of the reasons it cannot be 1 is because a 0 dimensional object cannot even assert its own existence.
Sure it can. This all sounds like you're completely mixing up everything at once. Just because a necessary axis of a 0-dimensional object doesn't exist, doesn't mean that the 0-dimensional object itself does not exist.