Colleges in Bangalore

You can visit this link:
https://www.shiksha.com/college-predictor.

Hello Shrishail
LLM. CLAT, AILET, LSAT, NEST are the exams through you can take admission. Generally universities offering LL.M. course conduct an Entrance Test for admission to the program. The application forms for this course are available in the month of April-May every year.
The entrance test paper contains around 150 questions which can be both objective and subjective in nature. The duration of the exam is 2 hours which includes negative marking for objective type questions.
Thank you.

Yes, you are eligible for B.A. Courses at SJC Bangalore. The eligibility criteria is given below. Eligibility : A candidate who has passed two year PUC examination of Karnataka State with any combination of optional subjects or any other examination from recognised Board/University or through National Institute of Open Schooling (NIOS) recognised as equivalent thereto shall be eligible for admission to B.A. Course.

Depends upon the eligibility criteria of the college.

Refer this link:
https://www.shiksha.com/university/azim-premji-university/azim-premji-university-national-entrance-test-net-exam

Yes, you can apply for IGNOU etc. if you want to.

Jain College is no longer offering MBA in logistics and SCM under Bharathidasan University, but it does offer the specialisation in CMS B-School.

It's good enough. The University has a good reputation and a large alumni base.

You have already done your Bachelor's. There's no point in doing it again. That too, in relatively the same programme with different specialisations. Try to get into a Masters program atleast.