Why is oxygen in a triplet state and what are the consequences? From Wikipedia here and here:
''Almost all molecules encountered in daily life exist in a singlet state, but molecular oxygen is an exception.''
''The unusual electron configuration prevents molecular oxygen from reacting directly with many other molecules, which are often in the singlet state. Triplet oxygen will, however, readily react with molecules in a doublet state, such as radicals, to form a new radical. ''
This wiki page is also relevant. Here is a picture (which I can't read).

How is this triplet state property quantitatively computed and why is it such an exceptional feature? 

How does the triplet state come about in oxygen? Don't more electrons in electron shell mean more complicated factoring into representations and therefore even more complicated states?

How does that impact the thermodynamical properties of the element and where is the thermodynamical difference (besides the different energy) to the singlet state? How can the reaction features be understood? 

Where does the energy difference of the two state come from. Taking a look at the periodic table of elements, has $\text S$ or $\text {Se}$ similar properties?  


Btw. I don't mind any math, but I'd probably need explanations for expressions like $\text O_2(b^1\Sigma_g^{+})$.
 A: There are lots of questions asked here, but I'll attempt to answer some of these...
Oxygen is found in the triplet state because the triplet state is most stable. This is a complex function of the properties of the atoms (e.g. charge and separation between atoms) and the electrons (e.g. number of electrons present, possible combinations of orbitals). The molecular orbitals given on the wiki page show three different states:
$^{3}\Sigma^{-}_{g}$: The ground triplet state
$^{1}\Delta_{g}$: The ground singlet state
$^{1}\Sigma^{+}_{g}$: An excited singlet state
These symbols are explained well at Wikipedia. In short, a triplet state has two electrons with parallel spins, for example the two red arrows pointing 'up' in the $^{3}\Sigma^{-}_{g}$ MO diagram.
If we investigate the energy of an oxygen molecule as a function of the distance between the oxygen atoms, we can uncover which of these states are most stable. An example of this is given here. It is evident that the $^{3}\Sigma^{-}_{g}$ state is lower than all other states, thus we expect triplet oxygen to be the most stable state. 
More complicated electronic configurations do exist, an example of these is given by the $^{1}\Sigma^{+}_{g}$ state. We can see here that it is even less stable than ground singlet ($^{1}\Delta_{g}$) oxygen. These variations in energy states arise because 'putting' electrons into different orbitals with different spins give varied 'goodness of overlap'.
Thermodynamic properties would be expected to be subtly different between electronic states. This is because the molecule would have different vibrational states which would 'translate' into different thermodynamics through statistical mechanics. I cannot find a source which directly compares such properties.
The reaction features are, of course, very different. For example, triplet oxygen will happily dissolve in water, but singlet will react with it. Singlet oxygen is often used when one wants to 'attack' double bonds. An explanation of why this is the case is very complex - but in short, is due to the quantum state of oxygen compared with the states of most molecules it attempts to react with.
