"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 points $x\in\mathring \alpha\cap\mathring \beta$ and $y\in \sigma$. Let $z$ be another point on the line $xy$ that lies in $\partial (\alpha\cap\beta)$. Denote by $\alpha'$ and $\beta'$ the smallest subsimplexes of $\alpha$ and $\beta$ that contain $z$. Observe that $\alpha'\cap\beta'=\varnothing$.

"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 $\alpha'\cap\beta'=\varnothing$.