Let $M,N$ be $C^1$-submanifolds of $\mathbb{R}^n$ and suppose that $\dim M < \dim N$.
Is it true that $M$ is a null set for $N$ (for the natural submanifold measure on $N$) ?
Is that result (if true) provable in a "elementary way" (i.e. not relying on Sard theorem for example) ?
I believe an elementary proof is possible, but you might need some further hypotheses - I'm not sure. Here's my thoughts (really just a long comment that won't fit in the box).
Suppose that $M$ and $N$ are regular surfaces in the sense that they are diffeomorphic images of parameterizations of $\mathbb R^k$ for some other $k$'s. Note that a Euclidean subspace of positive codimension has measure $0$ under the Lebesgue measure on $\mathbb R^n$. Now since diffeomorphisms are Lipschitz, and Lipschitz maps send null sets to null sets, then essentially $N$ is the Lipschitz image of a smaller dimensional Euclidean space, and thus is null.
You could probably do the same kind of thing on manifolds that aren't parameterized in this way by taking a finite atlas and stitching something like this together on charts.