BCA

No, because BCA is a UG course. And unfortunately, JNU conducts only PG courses now. You can do courses like MCA and M.Sc etc. from JNU, but only after completing your UG course.
Still, if you want to study a UG course from JNU, you must choose any foreign languages, because Jawaharlal Nehru University offers UG courses only in foreign languages right now. These UG courses include BA (Hons) in French, German, Korean, Persian, Arabic, Pashto, Chinese, Japanese, Russian and Spanish. Hope this helps!

Yes, It is one of the famous college. According to me, this college is good

BCA fee structure differ according to colleges or universities so please specify for which college you are enquiring after that we can only help you basic fee structure is INR. 1.5 Lacsto INR to 6 Lacs INR for three years, if you crack any good college entrance exam then you can do by less fee with the help of scholarship etc.

As there are many colleges in Maharashtra, I would suggest you that go for degrees from Pune University, as their degree are highly ranked in any of the state or abroad as well so go for it. And, also in Pune there are many top ranked colleges.

Some colleges will not have any problem if you studied maths in 12th or not for BCA course, but some colleges have the eligibility, criteria that student must have taken maths in class 12th. Therefore, it depends on the college you are applying for the course. Once you get admission in BCA you can then surely give NIMCET exam for MCA course in NITs.

Maths is there in computer science engineering in first 3-4 semesters. Physics and chemistry are taught during the 1st year only.

BCA course will begin in the August session.

Yes, you can if your 12th score is above 60%. First, you have to appear for SET (Symbiosis Entrance Test). There are three levels.
First is the entrance test, if you get cleared in that then PI-WAT(Private Interview and Written Ability Test) and if you score more than cut-out score, you will receive a mail that you have been selected for BCA.
Thank you.

You cannot get admission directly in the 2nd year after completing Dploma. You can get admission only in the 1st year.