How is dioxygen stabilised by coordination to haemoglobin?
I think you have a misconception. The binding of oxygen to haem and its release are processes that are entirely equilibrium-controlled by the equilibrium shown in $(1)$.

$$\ce{haem + O2 &lt;=&gt; haem{-}O2}\tag{1}$$

Meaning that wherever high partial oxygen pressures are observed, oxygen preferentially binds to haem but likewise it is released where low partial oxygen pressures are observed. Thereby, Le Chatelier’s principle nicely controls where oxygen is released from the red blood cells’ haem groups — it will always be in those areas where oxygen is required. Haem itself has evolutionally acquired the role of binding oxygen rather well but only to make sure that it actually reaches the distant parts of the body.

Oxygen in its $\ce{O2}$ modification is also not as bad as your initial post seems to assume. Yes it can react with quite a few substances and many of its reactions have a low activation barrier — but the concentrations in which it would be released into the blood stream ‘undesiredly’ are concentrations the body has learnt to cope with over millenia of evolution. So while minor damage may occur, most damage is quickly repaired by dedicated enzymes. Most importantly, any liberated oxygen will always be $\ce{O2}$ and never superoxide or peroxide.

Only while being bound to haem do oxygen’s (and iron’s) electronic properties change. On a microscopic level, the oxygen molecule causes a one-electron oxidation of iron converting itself to superoxide. This also causes a spin-flip in the iron centre which goes from high to low spin. The combined $\ce{\overset{\mathrm{+III}}{Fe}-O2}$ system can finally undergo antiferromagnetic coupling to give an overall observed diamagnetic ground state. Details can be found in this answer. However, while it is superoxide the $\ce{O2}$ moiety will never leave the iron centre — or, more precisely described, as soon as $\ce{O2}$ leaves, the entire process is reversed again. Thus, even though the bound state includes a superoxide ion no superoxide can ever be directly released into the surrounding blood stream.

$$\ce{[\overset{\mathrm{+II}}{Fe}(por)] + ^3O2 &lt;=&gt; [\overset{\mathrm{+III}}{Fe}(O2^{.-})(por)]}\tag{2}$$