India

There are many such colleges which have low fees but their infrastructure, faculty members, facilities etc. Are poor compare to others. However, you can choose that one which has an affordable price or you can go for scholarship exam too for financial support. Many colleges conduct not only entrance exam but scholarship also which is known as entrance cum scholarship exam. By getting higher marks you can get the seat as well as some scholarship also.
Best of luck.

Yes, you can apply for NPTA exam for BBA. It also offers distance learning mode. But for admission, you have to first clear the cut-off of the entrance exam.

Yes, your son can get a seat. But getting a seat in high demand courses like CSE and mechanical engineering is difficult.
Best of luck.

Unfortunately, no. The demand for CSE is higher and it requires at least 98.5% marks or above 150 marks in JEE. Try in other universities.
Good luck.

You can apply through another state quota. For that, you need to have a great score of more than 90 % to secure admission at MIT.
All the best.

Use the Andhra Pradesh-EAMCET 2018 College predictor at the link provided below:
Use your Andhra Pradesh-EAMCET 2018 rank to see where you can get admission. See the courses, colleges & rounds you may crack. The result is based on admissions data of Andhra Pradesh-EAMCET 2017. Get resources & insights on colleges & courses. https://www.shiksha.com/b-tech/resources/ap-eamcet-college-predictor.

Top executive MBA colleges in India accepting GMAT are mentioned at the link provided below. This ranking is based on the rankings published by Business Today. The mentioned exam cutoffs are for general category students.
Regards.
https://www.shiksha.com/business-management-studies/ranking/top-executive-mba-colleges-in-india-accepting-gmat/18-2-0-0-310.

DU JAT is conducted in pen and paper mode i.e. offline mode. Regards.

If you are a science student and wish to study BBA you can go for it. Check the college requirements for the above course.