If I have a continuous, and smooth curve $\mathcal{C}$, length $\ell$, in $\mathbb{R}^2$ and at each point on the curve I were to draw a line segment, length $d$, normal to the curve centered at the point; would the area covered by all the line segments be $d\cdot\ell$ provided that no two line segments intersect with each other?
Also: if this is true, can this be generalized to more dimensions?
No. One simple way to see this is to note that there are two choices of normal at any point, one on each side of the curve, and making the two choices won't lead to the same area in general.
To give a concrete example, let $C$ be a circle of radius $r$ (and let $d < r$ if you choose the inward-facing normal). Then the region swept out by the normal lines is an annulus, and you can compute its area for both the inward- and outward-facing normals; you'll see that you don't get the answer $d\ell$ in either case.