Do holes flow in a conductor? I want to if holes flow in a conductor and, if yes, why. What happens when holes reach the end of the semiconductor?
 A: There is no hole current in conductors because they have overlapping valence and conduction bands.
In conductors, electrons are loosely bound to the nucleus hence, can detach easily at room temperature. Also, a large number of free electrons thus, available are conduction electrons. When the covalent bond breaks, electrons are freed from the atom. The departure of an electron from the valence band creates a vacancy in the bond, this vacancy is known as a hole. This hole is captured by another free electron. Also, the energy level of free electrons corresponds to the conduction level, hence valance level and conduction level are overlapped. So, there are no holes in the conduction level to carry the hole current.
A: Since a hole is just an abstraction for an absent electron, when the hole enters a metal, this can be understood as an electron below the semiconductor's Fermi level vanishing from of the metal (so the current is understood to be carried by an electron on the metal side of things). Reversely, an electron appearing in the metal below to semiconductors Fermi level corresponds to a hole appearing in the semiconductor.
In addition to these one-particle processes, there elastic scattering processes (e.g. an electron being reflected on the boundary and creating an electron-hole pair).
On the technical level, the concept of holes amounts to a relabelling of operators: The electron creators respective destroyers below the Fermi level (which I understand to be the chemical potential at $T=0$) are relabelled as destroyers respective creators of holes. In this sense, you can easily rewrite the Hamiltonian of a metal in these new operators, and things will work out just fine. However, this rewriting is much less useful than for semi-conductors (e.g. because the chemical potential in a metal changes with temperature for fixed volume and particle number, while it stays in the band gap for a semiconductor).
Depending on the Fermi energies of the metal and the semiconductor and the doping of the semiconductor there are interesting effects on a metal-semicoductor junction, e.g. the Schottky effect.
