What exactly does it mean for the reaction of superoxide with non-radicals to be spin-forbidden?
What exactly does it mean for the reaction of superoxide with
  non-radicals to be spin-forbidden?


For clarity, let's start with a simpler case.  If we take a ground state singlet molecule (singlet means all electrons are paired) and we irradiate it with photons, photon absorption will excite an electron to an excited state.  Will the excited state be a singlet or triplet immediately after it is produced?  For the transition to the excited state to occur readily it must be spin-allowed.  That means that spin must be conserved.  If we started with a singlet, then our excited state must still be a singlet.  In that excited state the excited electron will still have its spin paired with the electron that was its partner that remains in a lower energy orbital.  Creation of a triplet state, where the excited electron and its ground state partner no longer have their spins paired is spin-forbidden and is therefor much less likely to occur.  Spin must be conserved.

Well rules are made to be broken.  What can we do to increase the probability of a singlet -> triplet conversion that is formally spin-forbidden. Spin-Orbit coupling (the coupling of the electron spin with the orbital angular momentum of non-circular orbits) is our way out. We need to find a spot where the Morse potential curves overlap for both the singlet and triplet. Then, if there is a large spin-orbit coupling interaction we can readily cross over from singlet to triplet, or back.  



Large atoms such as bromine and iodine, as well as transition metals with unpaired electrons have large spin-orbit couplings.  If such groups are attached to our molecule of interest, then spin need not be conserved and formally "spin-forbidden" reactions become more likely.


  Does the unpaired one have to bond with an unpaired one from another
  radical? 


Yes, if spin is to be conserved.


  What does this also say about reactions of the excited state
  of singlet oxygen (shown in the middle below) with non-radicals?


The singlet oxygen shown in the middle of your figure ia a singlet species and will react readily with other singlet (non-radical) molecules.


  is superoxide a doublet
  Yes superoxide is a doublet.  Ground state $\ce{O2}$ is a triplet - it has two unpaired electrons in energetically degenerate orbitals, if we add one electron to form superoxide radical anion, that electron will pair with one of those two electrons leaving only one unpaired electron.