CAT

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.

Coaching centers usually differ from city to city. Usually, in an area, local coaching centers are often better alternative than famous coaching centers. But as other mentors have mentioned, TIME is a good institute and has its branches in many cities.

More than 99.5 percentile in CAT is required to get the admission in IIM for MBA. You have a good score in 10th and 12th.

Oakbrook Business School, Ahmedabad is a good option. You can apply directly in the college by contacting the admin staff of the college.

Some common and important topics in logical reasoning are Coding and Decoding, Seating arrangements, Analogy, Letter Series, Number Series, Clocks and Calendars. These topics need good attention.

Not at all. Most people pursuing CAT have a gap in their education. It doesn't affect your CAT score.

You can apply for CAT/ other MBA exams before your graduation.

I think you can. Your PG degree must have mentioned somewhere that you are a Bachelor + Masters. So, you should be able to apply.