Utah

Yes, you can pursue bachelors as well as masters in Hospital Management. Yes, Techno India do offer Bachelor's in Hospital Management with 60 seats, full time 3-years course.

Hello Ajeet,
Please refer to this link to get more insights of the colleges and branches.
https://www.shiksha.com/b-tech/resources/wbjee-college-predictor
However, as per your rank, your chances are nil for getting CSE at Techno India.

To get into BCA at Techno India Salt lake, you need to have maths as a compulsory subject in class 12th.

All above colleges may be good. But for finding better college you have to visit the campus. Regarding this, I am suggesting you that you can visit one of college viz. Anand International College of Engineering, Jaipur and find differences in education system. This college provides good placements, exposing students in foreign for summer training, good infrastructure with Wi-Fi campus, digital class rooms, etc. So, you can visit the college campus and try to make your bright future.
All the best.

Kindly use sikha.com college comparator whose link has been provided below:
https://www.shiksha.com/resources/college-comparison

Which government college are you talking about? Obviously, government colleges are preferred over private colleges, but there are some cases where it is wise to choose better private colleges than to choose any average or below average government colleges.

You are eligible to get admission into Techno India.
Your rank is under the cut-off for Techno India.
You have to compromise with your branch.
Good luck.