Who is India's equivalent of the FAA? I frequently fly within India as part of my work and every time I am sitting in an airliner preparing to take off, I observe a lot of differences in each crew handling the take off or the ground crew procedures.

I would like to understand if there is any board that certifies these crews and airlines for safety, like the FAA for the US, if these information are released to the public, like a certificates that are issued for public information.
This regulatory oversight is performed by the Civil Aviation Authority of a country. In India, this function is performed by the Directorate General of Civil Aviation (DGCA-India). Typically certification information is available to the public, but only high level information will be given. 

Searching on the DGCA-India website I found this document that lays down the requirements on Cabin Crew Training.