The MO and unpaired electrons
Oxygen can exist in two states, the singlet and triplet state. The triplet state is the ground state whereas the singlet state is the higher energy state. In the triplet state the highest energy electrons are unpaired whereas in the singlet state the highest energy electrons are paired.
Now, take a look at the MO diagrams of both states in Wikipedia. You'll be able to tell that the singlet state does not obey Hund's first rule.
Now lets look at the Lewis diagram for a oxygen molecule:

As you can see, all of the electrons in this diagram are paired. The oxygen represented in the diagram above is an oxygen that has two bonds and two lone pairs. The only MO state that has lone pairs is the singlet state.
Now on to &quot;why does being paired in the Lewis structure equate to being paired in MO?&quot; The reason is that paired electrons in Lewis diagrams represent paired electrons. The concept of paired electrons does not depend whether you are using MO theory or Lewis diagrams. Paired electrons are electrons that are in the same orbital.
Now, the next question. &quot;Why are the name of MOs like σ and π? Doesn't σ and π get used when talking about σ and π bonds? Are σ orbitals and σ bonds correlated in anyway?&quot; Sigma bonds and sigma orbitals refer to the same thing. Molecular orbitals are a way to represent the bonds formed by two atoms.
Sidenote: Mulliken first proposed the sigma, pi, delta notation according to this journal article..