Indian Institute of Science

For UG courses, KVPY JEE and NEET are considered for admission.

It was around 240 (for general students) in previous years.

You may check the college website for their Contact information and Ask them about the Latest Cutoff Ranks.

You can only choose one major subject, although, in some of these institutes, you might be able to do a minor, whose criteria depend upon the institute.

Mathematics, along with physics and chemistry in class 12th, are compulsory for the BSc. course.

Unfortunately, no. The website states, "The candidates must have studied Physics. Chemistry and Mathematics as main subjects in their qualifying exam along with any other subjects. "
However, don't lose hope; there are other research-oriented colleges where you can still apply.

Yes, it is. Physics, Chemistry and mathematics are mandatory to apply for UG programme, even if you apply via NEET.

Unfortunately, no, only 1-year dropper students are allowed. That is, students who have passed 12th in the year of admission or the year before that.

That's not how it works. In IISER Pune and IISc, you can't choose your subject before admission. After two years in IISER Pune and three semesters in IISc, you can select your major subjects. Then you'll have to study the courses specified by the respective department; you can choose astrophysics if you have a course. Also, you can do your thesis in astrophysics if you want.

If you are planning to pursue research as a career, then no doubt these institutes are among the best in India. But in India, currently, earning money after B.Sc. and even M.Sc. is challenging as compared to MBBS. In the end, you should choose a degree of your choice because it is only then you would excel in your career. Also, both occupations are equally necessary; a country requires scientists and doctors to keep running the country, but unfortunately, only a few fractions of students take up research as compared to medicine.