What is the criteria for determining structure of a molecule?
No, not symmetry per se but generally the lowest energy arrangement of  the atoms.  What structure this is is not always obvious by just combining atoms according to the valency rules; if it were, experimental (analytical) methods would not be needed. This is particularly true of 'organic' molecules where one formula can give many similar structures, e.g. isomers, but also of the small molecules you list.

The rules we use are the result of numerous experiments over the last 150 years or so reacting one species with another and trying to make a consistent set of rules to explain things. But ultimately structural studies are needed, the definitive method is x-ray diffraction from a crystal of the material but of course this is a not always possible. In this case infra-red, Raman and NMR spectroscopies have been used as well as mass spectrometry.  These methods do not give complete structures directly (bond lengths and angles as does x-ray diffraction) but allow confirmation of one structure (as topology) over another, for example $\ce{N2O}$ is  NNO rather than  NON. More latterly computational methods are being used to predict structure but ultimately these will have to be backed up by experiments.