Ph.D

You can check with your MBA institute, whether online MBA degree issued by them is equivalent to MBA and approved by AIU. If it is AIU recognized and equivalent to MBA, then you can do Ph.D in management.

Yes, you can pursue MBA through distance mode. IGNOU and Welingkars are two great options. Ph.D can be specialized into above subjects.
Good luck.

PhD is awarded based on a research subject approved by university. Student can choose any subject in consultation with research guide. At the time of approving research proposal the Doctoral Committee of university may assign Course Work related to the thrust areas of research and research methodology to students.

It is no longer compulsory as per apex court decision in 2014. Good luck.

Hi,
There are many universities for the same. You may start with Indian Institute of Management, all IIMs offer FPM admission is based on CAT and full time, no course fee, you get stipend and admission is not easy, it is for 4 years duration. Other options are:
IITs
FMS Delhi
JBIMS
NMIMS
Symbiosis
TISS
XLRI

LLB is something, which you should learn and add to your portfolio. It will be always an added advantage.

If you are planning to take teaching as a career or want to become a management consultant with super specialization expertise, then PhD is a good option. But, if you are planning to be a corporate leader, Ph.D may not bring any benefits.