I have my interview in New Delhi on the 7th June. I gravely doubt they would schedule the interview earlier if asked... yes they ask for Police Certificates. I think for Indian nationals they ask for more than just the certificate... but I will have to go read the check list and let you know... not being an Indian myself I do not recall exactly what it is they want.
You need to submit originals... take a set of photocopies to the interview or they will keep the originals.