Are pi and sigma molecular orbitals for homonuclear diatomic molecules degenerate?

Answer 1

Generally it is not true.
Even for homonuclear diatomic molecules, please remember that the molecular orbits are constructed by mixing atomic orbits. When two identical atomic orbits mix, one of the resulting orbit energy is higher than the original and the other resulting orbit energy is lower than the original by the same energy gap, thus the total energy would not change. Hence there is no reason why pi and sigma molecular orbitals for homonuclear diatomic molecules would be degenerate.