Full Time B.Sc

Hello,
EIILM University, Jorethang, Sikkim, was established in 2006 under Sec.2 (f) of the UGC Act and established under ACT No.4, 2006 of government. Of Sikkim and the Act received the assent of the Governor on 24-03-2006 that was published under Government of Sikkim, Gazette Notification No. 28/LD/2006, dated 3rd April 2006. All the best.

Yes, you can do Master's degree regular after completing B.Sc.
Good luck.

Yes, but the university should be different.

Do regular B.Sc than the private one as it is preferable at the time of job.

Well, my advice would be even if you could, you shouldn't. The course is very vast with practicals which will overlap and exam datasheets which would overlap. Next thing it is not allowed anywhere in any university. You can read the eligibility criteria to get answer.

No, it is not possible. You have to choose either regular or private mode.

See, to be clear you can't manage both B.Sc and NEET together.
There is a saying that do not put each foot on different boats. So I suggest that do not mess up with 2 different courses. Just be clear about what is your goal, it is either B.Sc or getting MBBS through NEET. But remember one thing, you should be confident about what you want to choose. Your goal should be clear.