CBSE 12th

If you apply as a fresher (for first year), you will be eligible for almost every college in Bangalore, but if you want to apply for direct second year entry (lateral entry) then only few colleges might accept your admission due to state change.

Yes, you are. For more details check out the link below:
https://www.shiksha.com/b-tech/exams/imucet

Four months are more than enough for JEE Mains paper provided you are clear with the class 11th and 12th concepts. So, you need to just brush up what you have already studied. Don't go in for extra efforts or extra books at this 11th hour. Focus on self studies and try to remember the formulas and just solve the past JEE papers and MCQs more and more.

Go for MBA in SMU or IGNOU.

You can go for graduate/Diploma/certificate course according to your ability/choice. Go for a reputed institution.
All the best.

There is no option of BBA+ MBA integration after 12th board. It just offers a liberal studies and in 3rd year you have an option of choosing your major and minors according to your choice . The course is for 4 years.

Hi Devraj,
Firstly, you will have to pursue graduation in chemistry i.e. B.Sc chemistry (honors) from any of the reputed universities including Delhi University colleges. You can pursue your Masters from IITs by clearing IIT-JAM. There are other institutes as well like Tata Institute of Fundamental Research, Indian Institute of Science, National Institute of Science, Education and Research, Bhubaneswar.
All the very best for your future endeavors.