Why does H2 react with O2?
Indeed, much of the chemistry that takes place in the universe involves the breaking of covalent bonds and the formation of new covalent bonds. The key is that the energy of the products needs to be in a lower state than that of the reactants. This is the thermodynamic requirement anyway. Lets consider one of your examples:  

$$\ce{2H2(g) + O2(g) -&gt; 2H2O(g) + \pu{572kJ}}$$  

Although we had to break the covalent bonds of hydrogen and oxygen, the new bonds that were formed are even stronger, thus the reaction is thermodynamically favored.  

The complicating point of the above example is the fact that there is a large kinetic barrier to the reaction. This basically means that there is an intermediate product in the steps of this reaction that is energetically very high (unfavorable), so some catalyst is required (say, a spark) for this reaction to get over that energetic barrier and proceed on any reasonable time scale.  

Summary, TL;DR:
The breaking and creating of covalent bonds plays a major role in chemistry. The criteria for covalently bound compounds to participate in a chemical reaction is that the products have a lower energy state than than the reactants. If this is not the case, the reaction will not happen. Even if it is the case (that the products are lower energy / more stable than the reactants), there may be kinetic limitations requiring a catalyst in order for the reaction to proceed. But, no catalyst can make a reaction proceed if the above criteria is not met, that the products be more stable than the reactants.