Ph.D

Ys, but after your PG.

First of all, both the universities will ask for migration certificate at the time of Ph.D./LLB degree award and that time you can present only one. Secondly, Ph.D. is a research work which requires complete academic devotion and simultaneous pursuance of LL.B. may hamper that concentration. Lastly, being a Ph.D aspirant, if you can not find out UGC rules from their website then consider that you are not capable of research or Law work. So you can't do both LL. M. and Ph.D. part time.
Thank you.

Please mention the name of the university.

Go for Masters in BM from a good college in the field of your interest.

Here is the official link with relevant information:
http://www.iiserpune.ac.in/academics/int-phd-programme

Here is a link that can help you:
http://www.tifr.res.in/~dbs/students/int_phd.html

The duration can be anywhere between 2 and 5 years.

I am listing below do check the list:
Academy of Scientific and Innovative Research
Amity School of Communication, New Delhi Institute of Pharmaceutical Sciences and Research
Faculty of Education - Jamia Millia Islamia University
Indian Institute of Technology Delhi
Indira Gandhi Institute of Physical Education & Sports Sciences
Indira Gandhi National Open University.