Distance/Correspondence

Yes, you can give CAT exam, because CAT exam only consider approximately 10% from your previous education. Correspondence MBA doesn't have good value, but if you want to pursue, then you can.

Courses after MBA:
ISB Certificate Programme in Business Analytics (CBA)
Fellow Programmes in Management from Iims
NSE's Certification in Financial Markets
Executive MBA from a Reputed University/Institution
Doctor of Philosophy (Ph.D ) in Management
Chartered Financial Analyst (CFA)
Financial Risk Manager (FRM) Exam
Doctor of Business (DBA)
SAP course

Annamalai University is amongst the best universities in the country. It provides distance education in BBA. If you want to gain knowledge and succeed in your career, Annamalai University is a good option for you.

Entrepreneurship: Experience and Perspective
Creating Futures: Sustainable Enterprise and Innovation
Course: Lean Enterprise
These are the courses available for best distance learning.

The college for which you have asked doesn't provides this facility. But, there are lot of colleges/universities offering BCA from distance. Here is list of such colleges/universities. You can go through it.
https://www.shiksha.com/it-software/articles/popular-universities-offering-bca-mca-through-distance-education-blogId-14538

No, AICTE doesn't approve distance courses in engineering and technology. Regards.

As distance courses in engineering are not recognized so I would advise you to leave the idea of doing it. AICTE doesn't give approval for distance courses in engineering. Go for regular one. Regards.

You need to have 65% or 5.5 CGPA. You're not eligible for MBA.

MBA via the corespondance route is an option available for people who are employed. An MBA does give an opportunity for career growth. However an Exe. MBA from a top college would be a better option.