"Yes" for Question 1.

We need to show that interiors of distinct smplexes $\alpha$ and $\beta$ do not intersect.
Assume $\mathring \alpha\cap\mathring \beta\ne\varnothing$.
Then $\alpha$ and $\beta$ share a simplex, say $\sigma$.
Note that $\sigma\in\partial (\alpha\cap\beta)$.
Choose a point $z\in \alpha\cap\beta$ that maximize the distance to $\sigma$.
Denote by $\alpha'$ and $\beta'$ the smallest subsimplexes of $\alpha$ and $\beta$ that contain $z$.
Observe that $\alpha'\cap\beta'=\varnothing$.