Why H2 + O2 does not form H2O2
There is a lot more energy stored in the O-O bonding than you expect. It is very difficult to directly transform the O=O double bond to become an O-O single bond, which is what you propose doing.
Water has less energy per unit molecule than hydrogen peroxide. This is why burning hydrogen gas in oxygen most directly forms water vapor, rather than hydrogen peroxide.
Technically, for a stoichiometrically perfect representation, we prefer to say:
2 H2 + O2 -> 2 H2O
Combustion of hydrogen into water RELEASES energy.
Formation of hydrogen perioxide from hydrogen and oxygen REQUIRES INPUT OF energy.