Certificate

Yes, you can contact your board for the same.

No, it is needed during the admission process.

You only need sound practical knowledge to get a good job, there is no need for O level certificate.

Correspondence certificate has less value when compared to regular. Regular degree is given on attending regular classes and completing given tasks and on completion of the course certificate is awarded.
In correspondence degree, you just need to attend the exams and you need to prepare for the exams all by yourself. Henceforth, correspondence degree has less value compared to a regular degree.

For PGDM courses, there will be a GDPI round where they will evaluate you on the basis on your work ex if any, previous academic record, entrance exam score and your performance in GDPI itself. You will be selected for a course on the basis of these factors. So it basically depends all of these factors and not just on your profile. They will try to match you profile with the course that you have applied for and if matches then you stand a good chance to get selected.

It is not mandatory to have community certificate or caste certificate until you need a concession or scholarship based on your community.

You can actually check about IIHR, it is a reputed institute for HR certification courses. You have full time, par time and online mode of education there. It will help you to accomplish your goals.