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. (the website won't let me directly post images) Inversion map causes local intersecction
