In terms of highway classification, the recommended tagging scheme for India is documented on this Wiki page: India/Tags/Highway - OpenStreetMap Wiki
Essentially, ODRs are highway=tertiary, MDRs are highway=secondary, SHs are highway=primary and NHs are highway=trunk. Other village roads would be highway=unclassified.
On the topic of official government open map data, there is the PMGSY National GIS data which is compatible with the ODbL. There’s more details on this GitHub repo: GitHub - datameet/pmgsy-geosadak: PMGSY National GIS - Open Data
The raw data itself can be downloaded from the official page, though it may be geoblocked and accessible only from India: PMGSY National GIS - Open Data PMGSY GeoSadak
However, there are numerous data accuracy issues with the PMGSY data, either outdated or outright incorrect numbering/naming of SHs/MDRs/ODRs. There’s been some discussion about this on the OSM India Telegram group, some folks there may be able to provide more details about it: Telegram: Contact @OSMIndia