To add to what Babcock_Hall wrote, there are a lot of different bonding models. None of them are completely correct; all of them make certain approximations. The molecular orbital model basically says that bonds are made by mixing the atomic orbitals localized on the various nuclei to create new orbitals that span entire molecules (molecular orbitals). There are certain rules to determine how the atomic orbitals are mixed that basically come down to the fact that, to interact, the atomic orbitals have to have similar energy and appropriate symmetry. Moving on: the hybridization model is frequently used in organic chemistry treatments because it models bonding geometries well. This is not a part of molecular orbitals theory - in fact is a specific treatment of the valence bond model. The prime difference between molecular orbital theory and valence bond theory is that the former assumes that electrons in molecules are located in molecular orbitals that span the entire molecule, whereas valence bond theory assumes that electrons are localized in orbitals that are shared only between adjacent nuclei. Whereas molecular orbitals are created by mixing together atomic orbitals on separate but nearby nuclei, hybridized orbitals are new atomic orbitals that are created by mixing together multiple atomic orbitals on the same nucleus.
I know that this probably sounds confusing, and it doesn't really change things as far as fundamental rules like the way that electrons have to fill orbitals, but I just wanted to point it out because you're mentioning sp2 orbitals and sp3 orbitals, which have nothing to do with molecular orbital theory. If you need clarification on that, please let us know.
Getting back to molecular orbital theory, which is (with the exception of hybridization models) by far the more commonly encountered model in undergraduate courses: In most cases, the number of bonding electrons does exceed the number of antibonding electrons. As you've pointed out, if electrons fill orbitals that are lower energy first and bonding orbitals are always lower energy, there is no other way. At best, they would be equal, leading to a bond order of zero (as in a di-helium molecule). The only exception is if a molecule is excited such that a bonding orbital is promoted to an antibonding orbital - using light or electrical energy, say. In this case the number of electrons in antibonding orbitals can be greater for a short period of time, and this how molecules can photodissociate.
Regarding the diagram, the increasing density of states is complicated to explain without some linear algebra - but more or less it comes down to two factors. First, as mentioned above the number of molecular orbitals equals the number of input atomic orbitals. In an infinite crystal involving an infinite number of nuclei having an infinite number of atomic orbitals mixing together, there is an infinite number of molecular orbitals. Second, the density of states is the number of orbitals spread over a certain amount of "energy space". One thing you'll notice from the diagram is that the lowest energy bonding orbital and highest energy antibonding orbital are not infinitely separated, even as the number of nuclei become infinite. Rather, they asymptotically approach a finite limit. Since the number of states linearly increases but the amount of energy space reaches a finite limit, then the density which is the # of states divided by the amount of space becomes infinite as well. Importantly, the density of states here is an energy density, not a geometric density of the nuclei. The geometric arrangement of the nuclei affects the energy of interaction - i.e., how much the molecular orbitals raise or lower in energy compared to the starting atomic orbitals - but a similar effect will be observed regardless of their spatial density. I hope that makes sense.