Rajasthan

I would suggest you to go for Bachelors degree from UEM Jaipur, but there are some criteria to be fulfilled before the admission. So, check all the details needed from here:
https://www.shiksha.com/university/university-of-engineering-and-management-Jaipur-47406/courses/bba-bc.

There are a few colleges competing for best in Jaipur. I would suggest you to go for UEM, Jaipur and for this refer the following link:
https://www.shiksha.com/mba/course/masters-of-business-administration-university-of-engineering-and-management-Jaipur-236845.

In my opinion, mother's education hub is the best coaching institute in Jaipur.

No, you can not B.Ed after B.Tech.

https://www.shiksha.com/science/colleges/m-sc-mathematics-colleges-rajasthan.

Yes, it's good in terms of facilities, faculty member, placements and infra.

In Jaipur, you can opt for Government College:1) khaitan Polytechnic college(2) ITI College Bani Park, you can opt in any of these college, and if you want to take admission in private, there are many options of colleges. But, from my opinion you should opt for government colleges in both the cities Jaipur and Pune.