Can an organism process H₂O into H₂O₂?
Hydrogen peroxide as a reactive oxygen species (ROS) is a highly problematic molecule for the processes inside of cells (it can spontaneously oxidize and damage cellular components). With Catalase, organisms have a very effective enzyme which degrades the peroxide into water and oxygen.

However, although the molecule is problematic, it is still produced by organisms in traces. For its production by the NADPH oxidase, first NADPH is oxidized and superoxide is formed. The reaction looks like the following:

$NADPH + O_2 \rightarrow NADP^+ + H^+ + O_2^-$

In a second step the superoxide dismutase uses two molecules of superoxide and H+ to generate H2O2, the total reaction (which is divided into several steps) looks like this:

$2O_2^- + 2 H^+ \rightarrow H_2O_2 + O_2$

H2O2 is then usually degraded by the catalase (or used in immunological processes like the respiratory burst).