Why doesn't CaO decompose into Ca and O2?
Stability, the thermodynamic property, is defined as whether a decomposition reaction is exergonic or endergonic at a certain temperature. The reaction $\ce{CaO -&gt; Ca + 1/2 O2}$ is endergonic across a very large temperature range. But that doesn’t necessarily help you.

In that list of four, we have to clearly molecular compounds and two mostly ionic ones. We need different discussions for both. Basically, the molecules are unstable towards decomposition into $\ce{O2}$ and something ($\ce{H2O}$/$\ce{O2}$) because they contain energetic bonds whose replacement by an $\ce{O=O}$ bond is favourable.

For the ionic ones, we have a number of different discussions we can do. For one, the metallic partner is calcium versus mercury; while the latter is known since ancient times and pretty stable in liquid form the former is not: It will readily react to form salts. For another, the $\ce{Ca^2+}$ ion is rather small and thus a hard Lewis acid according to Pearson’s concept. $\ce{Hg^2+}$ is much larger and thus a much softer acid. Therefore, the adduct of $\ce{Ca^2+}$ and $\ce{O^2-}$ is predicted to be more stable (hard–hard interaction) than that of $\ce{Hg^2+}$ with $\ce{O^2-}$. Finally, we can also consider heats of formation which should (I unfortunately don’t have values) be much more negative in the case of $\ce{CaO}$ than for $\ce{HgO}$ — therefore, the latter can be decomposed with enough heat due to $\Delta G = \Delta H - T \Delta S$ while the former cannot.