Do all flames require the same minimum concentration of oxygen to sustain themsleves?
No, different substances will cease combustion at different concentrations of oxygen. The concentration needed can be described by the limiting oxygen index. This is the reason that the use of pure compressed oxygen can be so hazardous; not only will materials burn that would not have burned before, but now materials that do burn require a lower temperature/energy input to ignite.

You can understand this conceptually when you consider where the heat goes. On combustion in air some of the heat must be used to heat the $\pu{79\!\%}$ of the air that isn't oxygen and that heat is wasted in the inert gas and the heat is diluted. That means only ~$\pu{20\!\%}$ of the heat goes into activation energy to propogate the combustion. When the oxygen is more concentrated, less heat goes into inert gases and more goes into oxygen to propagate the combustion and vise-versa.