Your marked-off sidewalk is not a dimension. But it could be considered a basis vector. You can have ten sidewalks running in ten different directions but you do not have ten dimensions as a result. A maximum of two of them may be taken as basis vectors because the ground the sidewalks are built on is two dimensional. The 2D ground exists as a manifold in our 3D space. It is a manifold because there are certain restrictions that limit some phenomena and makes them stay on the ground. For example, sidewalks are built on the ground and can't be marked off to measure altitude. Of course you could use a ladder to form a third, orthogonal, basis vector and thus form a coordinate system over three dimensional space.
I agree that you can see the ladders and sidewalks, but they are not dimensions. I maintain that you cannot see dimensions.
What you can do is to align your line-of-sight along any of the three directions of any three basis vectors you choose, as well as along any direction which is a linear combination of these basis vectors. You can look in any of the directions in three dimensions, but you cannot see the dimensions themselves. The dimensions are simply numbers that characterize one aspect of the space you are talking about.
The point I am getting at is that there is no reason why our 3D space couldn't be a manifold in a 4D (or higher dimensional) space with the extra dimensions physically inaccessible, "invisible", and undetectable by any structure (eyeballs, brains, telescopes, cameras, etc.) available to humans. After all, as Alex points out, everything we have at our disposal to "see" with is made of 3D atoms and molecules. Since all of those things are confined to our 3D manifold, just as the sidewalks are confined to the 2D manifold, they cannot be "aimed" or aligned with any direction which is not a linear combination of any accessible set of three basis vectors over our manifold.
If such extra dimensions actually exist, they would provide "room" for many things that are posited by many people but are difficult to explain otherwise. Some examples are, the curvature of space in response to mass; many worlds; all kinds of claims for spiritual entities; and your "cosmic mind" (forgive me but I can't remember what you called your notion of a super mind existing somehow in the dark matter). In fact, in this last example, extra spatial dimensions would not only provide the "room" for such a super mind, but would provide an enormous increment in the possible complexity for such a thing, and it would also allow nearby, immediate, and undetectable access to everything in our 3D manifold.
I just think that possibility is something to think about when you speculate on what else might be going on in reality.