Science

Biology at 10+2 stage is not compulsory for an undergraduate degree in Psychology. The basic eligibility for admission to BA Psychology programme is a pass in intermediate or 10+2 examination from a recognised board or university. However, some colleges may require prospective students to have studied psychology as a subject at the intermediate level.

Courses after 12th Science with PCB:
MBBS.
B.A. M. S. (Ayurvedic)
BHMS (Homoeopathy)
BUMS (Unani)
BDS
Bachelor of Veterinary Science & Animal Husbandry BVSC AH)
Bachelor of Naturopathy & Yogic Science (BNYS)
Bachelor of Physiotherapy.

No, you cannot do B.Sc.

Dear,
You should go for BA. Here you can find the subjects like history, economics, psychology and many others which will be helpful in your preparation.

You should be M.Sc in mathematics.

https://www.shiksha.com/science/colleges/m-sc-mathematics-colleges-rajasthan.

Please refer www.ssbcrack.com for getting knowledge of defense related post.

You can apply for IISER after class 12th. The ways to apt for IISER is given in below links:
https://www.iiseradmission.in/?page_id=8.

Since you have not got admission, you must try again. You can go for M.Sc Physics or M.Sc Astronomical physics. You can study MS or post-graduation from abroad by giving some exams like GRE or IELTS. These are entrance exams for admission in post-graduation.

Yes, you can appear for NDA exam in class 11th and the first post rank given is Lieutenant in officers.