Does a catalyst lower the energy required for decomposition? Exact Reaction - H2O --> H2 + O2

No it only increases the rate of forward or backward reaction

Decomposition reactions are reactions that break apart a single reactant into elements or simpler compounds. In many cases, energy input or a catalyst is required for decomposition reactions to occur. 

This is a redox reaction. 
Oxygen is both oxidized and reduced.
O in H2O2 has an oxidation number of -1.
The oxidation number of O goes to -2 in water and 0 in O2.