M.Sc

Yes, you can definitely apply for the M.Sc in biomedical science after doing B.Sc in medical Biotechnology if you wish to pursue the same.

Delhi University releases a cut-off list and if your name is mentioned in the list then it means you are eligible to apply for the admission process, otherwise, wait for the next cut-off list.
Good luck.

You can apply for admission only if you fill the registration form, so you should fill the same and also visit the university website for the eligibility criteria.

One of the ways to get into MS course in India is through GMAT. You can apply for GMAT. It's an international level exam conducted across the world. Many top institutes of India accept GMAT score.

You have the option of doing an integrated course, but not dual degree courses. Maybe that would be on offer in the distant future.

https://www.shiksha.com/b-tech/articles/iit-courses-blogId-19985
https://www.shiksha.com/b-tech/articles/iits-in-india-blogId-12884
check out these links for information.

Entrance test pattern for M.Sc in Counselling Psychology.
In Part A, qQuestions will be asked from the following topics-
1. General Aptitude
Logical and analytical reasoning, Decision making, Linguistic skills (30 questions)
2. Basic Psychology
History, Schools of thought in Psychology; Perception, Motivation, Personality, Learning, Social psychology (15 questions)
3. Current Affairs
General knowledge/ awareness about world around us (5 questions)
4. Research Aptitude
Basics of research methodology (5 questions)
5. Ethical concerns (5 questions)
Part B consists of Counselling Psychology.

* This is the average fees (per year) INR 10,000-50,000.
Visit the link:
https://www.amrita.edu/admissions/bsc-hons-agriculture-2019/fee.

These are the entrance exams:
MBA
CAT
MAT
XAT
CMAT
MH-CET
ATMA.