B.Sc

You can go for:
1) Government job.
2) You can give entrance exams like CGL and UPSC.
3) You can do Master of Science.
4) Library Science.
Since you are a Bachelor of Science student, various opportunities are waiting for you.
Go ahead.

Yes, you will be deemed eligible upon clearing the CAT and IRMASAT cut-off (conducted by the college). The cut-offs are determined by the college based on applications.

See, MBA is a stream which in today's world consists of various streams. I too was in the paramedical stream and now doing MBA. If you want to earn well and you are fluent in English then only go for MBA.

You are not eligible. You are not having biology as your core subject.

B.Sc Eligibility criteria in terms of mandatory science subjects (PCM/ PCB/ PCMB) that candidates need to have in Class XII to secure admission in different B.Sc. Courses (Honours/ General) is mentioned at the link provided below:
https://www.shiksha.com/science/articles/what-is-bsc-course-all-about

Basic eligibility for admission to Tolani Maritime Institute for Diploma in Nautical Science is that candidate should have secured 60% marks in class 12 with mandatory subjects : Physics, Chemistry, Mathematics
Candidate must have secured minimum 60% marks in mandatory subjects and must have studied

Yes you should write or you can take admission directly during the admissions process. Many seats are left so to fill those seats they take some of students but for that you have to book the slot earlier.

Bachelor of Science Jobs:
Jobs for students of B.Sc are just not restricted to areas of science. They can get jobs with other sectors too. With some training and expertise in a certain field, one can get work with a wide range of non-scientific arenas too. Few of the employment areas after B.Sc area:

Regarding qualification for a teacher, you can go for either B.Sc or B.Tech. But if you want a career in teaching field then only you should go for B.Sc followed M.Sc, NET or B.Ed.

No, it's not compulsory.