B.Sc

Yes, you can take admission in private universities or colleges. Even, you can write some entrance test for some private institutions.

The average course fees for the Bachelor of Science [B.Sc.] (Agriculture) course range from INR 52,000 to 1 lakh per annum. This course fee varies based on the government and management quota as well as the prestige and position of any university or institute.
The best and most accurate way to know the fees structure is to visit the official website of the college or university. If the cutoff has already been issued, then there you will get the complete list of fees structure for the respective course.
Hope it helps.

B.Sc. Nursing is a course of span 3 years and some of colleges for B.Sc. Nursing in Bhopal are RKDF college of Nursing, Mansarovar Nursing college, Rabindranath Tagore University and many more.

Firstly you must have 10+2 or equivalent degree with minimum 50% marks in Physics Chemistry English and Biology or Mathematics but Biology is compulsory. Then admission is done on the basis of marks secured in entrance exam.

There are a lot of colleges in Mumbai for B.Sc. Some of them are St. Xavier's College, University of Mumbai, Bhavan's college, K J Somaiya college of arts and commerce. It is a three year long course.

Yes, need to quit from B.Sc. If you want to do B.Tech through JEE Main.

No, it is not available.
Jadavpur University B.Sc Course Details
It is a three-year full-time course divided into 6 semesters. The institute provides B.Sc degree across five specializations:
Physics
Chemistry
Mathematics
Geological Sciences
Geography
Jadavpur University B.Sc Admission Guidelines:
Applicants will be required to appear for the admission test conducted by the university.
After declaration of the result, a merit list will be released by the university for further selection.
On the basis of the merit list, selected candidates are called for the interview for final admission.

B.Sc. nursing is a course of 4 year. In Rajeev College of nursing you have to pay around INR 70k per year and you need 10+2 or equivalent degree with minimum of 50% marks in Physics, Chemistry, Biology and English.