I don't think there is a problem here.
S + H2
S=O + H2
S=O + H2
How to avoid hydrogen peroxide in product? Stop the addition. Dimethyl sulfide is a pretty good reductant. I don't think you need to do anything else.
I'm not seeing this. What is the mechanism here? You can make the anion of hydrogen peroxide without decomposition. If you want to do a conjugate addition to an enone, use hydrogen peroxide and (catalytic amount) base.
I can write a balanced equation for the formation of oxygen and water without base. Therefore, a catalyst only need to change the reaction path, not the stoichiometry. As described, this is a stoichiometric reaction, "If you stop the addition of LiOH the reaction stops." I do not understand what is being oxidized.
If you do a Payne oxidation, if the reaction is basic, then oxygen is released.
RCN + H2
+ O (bracketed)
If not basic, the peroxy imidate can be intercepted to form an epoxide.