Manipal University, Jaipur

Kindly visit this link:
https://manipal.edu/mu/admission.html

Why are you so much confident that you will get your branch in such type of rank? You can't get admission, because your rank is too high. Even CSE lasts for about 18000 rank only.

Yes, Manipal University offers BJMC in private mode. The fee may be high.

It's good, but I would suggest you choose any other college for BA LLB as there more of the mass bunks and rescheduling of the class than studies here because most of the law students are localities and now you can imagine the rest but in terms of studies, the teachers are good and everything, but the surrounding atmosphere is somewhat not so pleasant, it's not good only for law students but for all.

Manipal University is far better than Amity University. Manipal provides better environment and infrastructure and helps in building career in the respective field.

No, CSE is the best stream in Manipal, Jaipur but if you are interested in mechatronics, then you can go with that. It's good but not the best.

MIT have its own entrance to join their courses. If you have JEE Score, it will add value to your application over there.

Yes, you need maths if you want to pursue B.Com (Hons) from Manipal University, Jaipur.

It will depend on cutoff for this year but for an idea, you can go through the previous year's cutoff. You may get branch according to your interest in Manipal University.
According to last year cutoff you may get admission in manipal university you can try.
Good luck