CAT

Yes, you are eligible, given that you have scored a minimum of 50% in your UG (45% in case of the reserved class).

Yes, you are eligible to write CAT exam. You can apply for the same.
Good luck.

These are do or die chapters in mathematics. Limits, continuity & differentiability. Application of derivatives. Definite integration.
3-D geometry. Sequence & Series. Probability. Straight lines. Circles. Matrices. Complex numbers.

You must complete your BBA before the academic session of management begins. If you just reached final year then you are not eligible for 2018 exam.

Hello Farooq, eligibility for GIM PG health management is a valid score in CAT, GMAT, CMAT exams only. 95+ cutoff in CAT, 99+ cutoff in CMAT have fair chances to get admission.
All the best for your future.

You will need 99+ percentile in CAT to get FMS. FMS is usually a top choice just after IIMs. Hence you need to score higher.

You need to have 65% in CAT or MAT for admission.

Yes, after the completion of your BBA course, you can apply for IMT-Hyderabad through CAT. But as this year, the admissions process is closed you can apply for 2019-20 batch.

Your academics scores are decent.
All the IIM A and B have a different CAT cut-off.
Like IIM-A's minimum 80%ile and the sectional cut-off is also given weightage along with class 10th, 12th and graduation marks, and work exp. Similarly, in IIM-Bangalore and Calcutta, the minimum is 90%ile and other factors matter too.

Of course, that would be very helpful for your career. Also, I should suggest pursuing PGDM if not MBA. Since you already have the experience, the MBA degree will open more opportunities for you.