B2 is a good call! It does indeed seem to be a case where 2 "half pi" bonds are stronger that a single sigma bond. However B2 is paramagnetic so high spin could play a part in that.
Yes, the back-bonding in complexes of ethylene with metal ions would certainly be candidates for pi bonds - though the forward donation part of the bonding could be construed as 2 sigma bonds. This is territory where things are not so clean-cut!
When you think about it, any atom with, say, p orbitals available to form pi bonds would also be able to form sigma bonds with them; and that sigma would be more energetically favoured. So we would have to rely on steric constraints preventing sigma bond formation. It is difficult to imagine a...
If v is volume per mole (v=V/n) then the second equation is identical to the first if we divide both sides by n.
As for the third, different gases have different molecular weights so if m is a mass one would have to have a different R for every gas to make it true.