Have you considered the simplicial 3-manifold cases? For Question 1: I think an almost-embedding can also be a 3D simplicial complex (tetrahedral mesh) mapped to $\mathcal{R}^3$ (for example, an identity map). For Question 2: If you have an extra condition of $\nabla f > 0$, I think the hypothesis holds in this case. Otherwise, you might have degenerate cases or flipped cases, like this $\mathcal{R}^2 \mapsto \mathcal{R}^2 \times \{0\}$ case shown in the figure. Because the bijectivity of already adjoint simplexes is not controlled by the definition. [Inversion map causes local intersecction][1] [1]: https://i.sstatic.net/cNkrf.png