Science

You should consult with admission authorities of your institute whether you can get lateral entry. Having a good academic record will also help to secure admission. You should block a seat in the college you want to study before giving up your seat in your current college. Institutes normally allow lateral transfers under special circumstances like medical issues, shifting base from one place to another, etc.

Please refer to the link provided below for details regarding various entrance exams for M.Sc programs in various colleges in India.
https://www.shiksha.com/science/articles/top-bsc-msc-entrance-exams-in-india-blogId-13484

Eligibility for MBA program: The prospective applicant must satisfy one of the following eligibility criteria for admissions to the MBA program. A graduate degree under 10+2+3 or 10+2+4 pattern under any discipline, securing at least 50% marks in aggregate, from a recognized university.
However, different universities have different admission criteria and you should check with the admission authorities of the target institute for eligibility criteria for a course in their MBA program.

Dear,
As IGNOU is a central university and recognized by UGC, its Bachelor's degree is equivalent to other UGC recognized B.Scc degrees with the same combination. You just need to check whether three years Bachelor's degree is validated by their university or not. I answered this from my common sense without having any first hand information.
Best wishes.

Dear,
Based on your query, I would like to suggest the following options:
1. You can go with B.Sc (Hons) in your favourite subject followed by M.Sc.
2. You may think for BCA follows by MCA if interested in the software domain.
3. If you are interested in the Engineering domain, think for B.Tech where your math skill is very much useful.
There are so many wings to study, go ahead based on your interest, skill and industry requirement.
Best wishes.

Yes, a graduate from UGC recognized university can apply for civil service examination.