BBA

Amity University is a great option for you. With a widespread career opportunities and a great college environment, Amity University Lucknow Campus is a great option for you.

It's good that you want to do MBA from good college but it's not compulsory that you have to do it from Jaipur. There are so many colleges in other cities and Jaipur is also a big city. So, you can get a good colleges there also.

Yes, you are eligible.

CIMS has totally 180 intake. 90 seats Management & 90 seat Government. Students who have written MAT/KMAT/CMAT/ATMA/CAT can apply for Management quota and students applying for Government quota, should come through PGCET only.

Hi,
Admission is based on marks obtained in 10+2.
https://www.shiksha.com/bba/course/bachelor-of-business-administration-gitam-hyderabad-business-school-patancheru-276137

Hi, Please refer to the link below:
https://www.shiksha.com/accounting-commerce/articles/course-options-after-b-com-blogId-12939
https://www.shiksha.com/accounting-commerce/articles/careers-in-commerce-attracting-more-and-more-students-blogId-6805.

There's not any more course available. Only IMPAT.

Yes, you can pursue BBA after you puc Science and you even have an advantage of having studied Science in your puc. You just have to cope up with commerce subjects but that won't be a big problem for a Science student.
So, good luck.