National Law School of India University, Bangalore

No, NLSIU offers BA LLB only.

I think you should opt for NUJS(regular) because when you opt for distance programme you miss out on many things including classrooms discussion, faculty, debating etc. In NUJS, MBL course has produced many good brains which later came up with many good startups. If you want to prepare any exams then you can opt NLSIU (distance) programme, otherwise, I would recommend NUJS (regular) programme.

Yes, there is an NRI quota where 5 students are admitted every year. For details on fee, please visit the following link: https://www.nls.ac.in/index.php?option=com_content&view=article&id=43&Itemid=28.

I am not sure about the Diploma but NLSIU regularly conducts certificate course in Medical Law and Ethics. Refer to distance education department website for more details. (http://ded.nls.ac.in/).

No, the eligibility is the bachelor's degree. For more details refer to the distance education department website of NLSIU. (http://ded.nls.ac.in/).

Yes, you can apply for DU LLB after completing your graduation.
DU conducts it's entrance exam every year in a month of July.