Question

Which properties of carbonates, hydrogencarbonates, oxides of carbons and carbides put them under the study if inorganic chemistry and not organic chemistry even though they are also carbon compounds??

Answer 1

Organic chemistry is the branch of science to the study of carbon containing compounds - so all organic compounds contain carbon. But the converse is not true:
Some of the carbon containing compound are not studied under organic chemistry. This applies to the
CO₂, CO, CN^- , C₂^2-, CO₃​^2-, HCO₃^-, etc.

The definition of organic compounds is probably better expressed as: carbon containing in which there is C-C and C-H covalent bond. Inorganic carbon compound may contain either C-H or C-C bonds, but not both. That is why ​CO₂, CO, CN^- , C₂^2-, CO₃​^2-, HCO₃^-, etc. ar studied in inorganic chemistry.