It sounds to me like what you're trying to get at is the idea of a Normed vector space. Logically, the space of points in a universe is a separate concept from the space of distances among those points; in principle you could have two universes with the same points but where the distance from A to B is different in universe 1 from what it is in universe 2, and the shortest path from A to B takes you through point C in one universe but not in the other. So when you wake up and find yourself in an N-dimensional universe, you still have to make observations in order to figure out what geometry you're in.
To say "there should only be up, down, left and right directions" is to say that intuitively you expect to find yourself in a universe with a "p-norm" where p=1. What it means to say that there is a diagonal direction is simply to say that this intuition is wrong for the universe in question -- that in fact the local p is some other value, typically 2 -- you're in a universe where the Pythagorean Theorem applies: distance is (up2 + left2)1/2. You could have found yourself in a universe with a different p. I spend my working life in the universe you expect, where p=1 and distance is (up1 + left1)1/1, i.e., up plus left. There are no diagonal directions -- to get a wire from transistor A to transistor B you have to go only up, down, left and right, because the routing software won't allow you to do otherwise. (It's written that way because to try to go diagonally will probably block a hundred other wires you haven't laid out yet.)
The point is, to say "So in physics and mathematics, superpositions are allowed." isn't quite right. In mathematics, whether superpositions are allowed is up to you. You can allow them or not, and then study the implications of your choice. In physics, whether they're allowed is up to the world you find yourself in. You can't rely on intuition; you have to observe the world and see what its p is.
I guess this all comes down to the strangeness of infinity. I was thinking about a completely dense plane equivalent to R2. So if you were to travel diagonally, you would never come to a gap in the fabric of the continuum of the plane, and I realized that there is no shape that allows this and still fills in the entire plain.
Now I just realised that they are dimensionless points, and somehow if you put enough of them together, specifically with the density of the real number line, the points become an area.
I just have to deal with how amazing and strange infinity is.
You seem to investigate"how objects have position in space", "how points are connected" and "which moves are allowed".
Neither of which has any obvious connection to the naive notion of orthogonal coordinate systems.
Instead there should be a lot of insight by studying quantum relativity physics (and maybe string theory).