LLB

You cannot pursue LLB. To do LLB, you need to complete your graduation first. The second option is 5 years LLB after 12th
NUJS-Kolkata is the best institute but you need to clear CLAT. There are many institutes under Kolkata University for which you can apply but most of them offer 3-year law course, so you need to complete the graduation.

BBA will prepare you to pursue something the field of management like MBA, etc. If not interested in MBA, then you could opt for CA, CS, CFP, ICWA, M.Com, etc.
With a BBA LLB degree, you can work as legal advisor, law reporter, legal assistant, lawyer, etc. You can opt for courses on corporate law, acquisitions, mergers, etc. after BBA LLB to find better career prospects.

The cut off of a college changes every year. For this kindly contact college authorities or check their last result on the website or the internet.

Both the fields have same demand but most of the students are choosing B.Tech because getting job is easy with B.Tech. BA LLB is also very as very few students are pursuing and you can start your own practice but you have to work hard in law field to get success.

National Law Universities established by Government of India are the best option for doing 5 years integrated programme of law. If you can crack CLAT then it's the best option to become a lawyer. It will be very good for your career and above it, you will save one year of education.

Both stream offer a variety of career options, however, they are very different in the approach so you need to decide which is on the priority.

You can check:
https://www.shiksha.com/law/articles/bhu-uet-llb-exam-previous-year-question-papers-blogId-13196
https://www.shiksha.com/law/exams/du-llb-entrance-exam

CMR is third best option for doing 5 years integrated law degree after NLU and Christ University. You may refer to following link for law degree courses ranking in Bangalore: https://www.shiksha.com/law/ranking/top-law-colleges-in-bangalore/56-2-0-278-0

National Law Universities are the best institutes for five years LLB degree courses. If you are able to crack CLAT then it's the best option to become a lawyer. You may refer to following link for list of top law colleges in India: https://www.shiksha.com/law/ranking/top-law-colleges-in-india/56-2-0-0-0.
You may select your college on ranking of college not on whether it's government or private university.