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][1]


  [1]: https://i.sstatic.net/cNkrf.png