Exam

Yes the do conduct entrance exams.

You can go through the link given below-
https://www.shiksha.com/college/indian-institute-of-management-Tiruchirappalli-Chennai-campus-ekkatuthangal-49131/admission.

Yes, you do this as a private student not a regular. The other option is NIOS on-demand exam.

For your OBC category up to 17k to 29k rank is required to get into Tezpur university for B.Tech in mechanical engineering.

No, you don't need to pay it again. Authority will share relevant details with you.

Hi, First of all, don't worry about it, By experience, I want to share a top-secret with you JEE is not the exam that judges your academics, It judges you more on the basis of how you handle pressure and Engineering is also the field of how you handle pressure, during Engineering, they teach you a lot about pressure handling and thus companies need engineering so that they have people who can handle pressures and complete the work within the pressure at a much faster rate. So, Just Prepare yourself well and just make yourself less stressed and maintain good consistency.

No. It completely depends on you. Just do your hard work and you can achieve anything!

Hi
If you are an GATE aspirant, then there are many courses as well as job opportunities you can go for if your rank is below 100 (open category) and rank below 1200 and 2000 (SC and ST ) category then you can go for various PSU if you score good you can even go for M.Tech from IITS, NITS, IIITS etc. in various courses.

It depends upon college to college. Check the cutoff marks/percentile of the college you wish to go for.

CAT is for MBA and not for BBA.