How can we exclude vertices of a compact polyhedron and represent it as a convex hull?

Question

It is known that every compact (closed and bounded) polyhedron $P$ can be written as a convex hull of finitely many points, i.e., $\text{conv}\{x_1, \dots, x_m\}$.

Questions

1. How can I make sure that, I am always able to find these $m$ points?
2. Are these points vertices of the polyhedron?
3. When we find these $m$ points, are they unique?
4. Under what condition it true that $P \backslash \{x_1, \dots, x_m\}=\{\theta_1x_1+\dots+\theta_mx_m \mid\theta_i>0, \sum_i^m \theta_i=1, i=1,\dots,m\}$?

My thoughts:

Since $P$ is a polyhedron, one needs finitely many inequalities and equalities to represent it. We should be able to come up with some methods to extract vertices from them.