Why does oxidative phosphorylation require complexes 2 and 3?
The numbering of the complexes in the respiratory chain is confusing, as it creates the impression that the chain is structured like

Complex I $\rightarrow$ Complex II $\rightarrow$ Complex III $\rightarrow$ Complex IV

... but this is wrong. There are in fact several ways of feeding electrons into the respiratory chain. They all begin with one enzyme E that oxidizes an electron carrier in the mitochondrial matrix, such as NADH, FADH2, or ETF (electron transfer flavoprotein) and transfers electrons to coenzyme Q, which is then processed by Complex III, after which electrons are handed on to Complex IV. Both complex III and IV pump protons. But there are several options for the first enzyme E:


Complex I (NADH dehydrogenase). This handles all electrons from NADH (from many sources in oxidative metabolism) and is typically the major source of electrons in the chain. It is the only enzyme E that also pumps protons itself.
Complex II (succinate dehydrogenase) accepts electrons from succinate (a step in the TCA cycle).
ETF dehydrogenase accepts electrons from reduced ETF, notably from fatty acid oxidation.
Glycerol-3-phosphate dehydrogenase accepts electrons from cytosolic NADH (from glycolysis) via the glycorol-3-phosphate shuttle.


There are more alternatives, but I think these are the most important ones. So you are absolutely correct that when NADH is the substrate, Complex II is not involved. There is in fact not one respiratory chain, but several possible chains with different starting points E,

E $\rightarrow$ Complex III $\rightarrow$ Complex IV

You might think of this as a "fan in" structure where the various enzymes E all converge on Complex III. (I don't have the energy to draw this :)

As for the proton pumping at Complex III, it is not the case that the H$^+$ pumped are exactly those derived from NADH --- the system is more complicated than that. The respiratory chain(s) convert chemical energy from the redox reactions into potential energy (a proton gradient), but most of the H$^+$ pumped are obtained from the mitochondrial matrix. The exact number of protons varies depending on the substrate: for example, the 2 electrons donated by NADH at Complex I is sufficient to pump up to 10 protons. Actually, the stoichiometry is not fully understood, and there isn't really an exact integer number of H+ pumped, because this pumping is an imperfect physical process --- there is leakage, protins slipping back into the matrix etc.