Jain Deemed-to-be University, Bangalore

I don't know about Jain University, Bangalore so I cannot comment on this. But here in UPES, BBA course is above average.
If you compare placements with the fee, you will not be much happy. But I know some seniors here who have started their own start-ups.
So you will learn a lot of things here.

Go for Jain University without any second thought. Had you opted for B.Tech, UPES would have stood out.

Yes, you can still apply for it. Admission is on. Go to https://www.jainuniversity.ac.in/Admissions-in-Top-Colleges-of-Bangalore-India-offered-by-Jain-University.php.
All the best.

Yes, you are eligible for that.

Yes, you can get admission as the minimum eligibility criteria for the same is pass in the 12th standard from recognized boards like CBSE/ISC/PUC or equivalent board.