All positive integers appear as orders of cyclic subgroups of orthogonal groups, so perhaps Question 1 should be made more fine-grained. For $n=4$, you can make a list of finite order subgroups using the product decomposition of $Spin(4)$ and the $n=3$ case.

@Schemer1 The condition missing from the question is that the completion must be at the identity. In my answer, I give an example of what happens when completing at the non-identity point $1 \in \mathbb{G}_a$. This is why I chose $\mathbb{G}_a$ instead of $\mathbb{G}_m$.

@DanielSebald I'm pretty sure the convex hull has many additional edges. To construct the graph, I am restricting my attention to those edges that come from certain central angles.

@YuanYang Yes, I think it would be more precise to say that after pulling a section back to the normalization of the curve, the residues at the preimages of the node add to zero.