CBSE 12th

Hi Anirudha,
There are many competitive exams one can write such as JEE Mains, JEE Advanced, BITSAT, VITEEE, NATA, NDA, Indian Maritime University Common Entrance Test, Indian Army Technical Entry Scheme, NIFT and so on depending on one's own interests. One can also attempt the state/college common entrance tests.

Please refer to the link provided below for details regarding admissions in B.Sc. course after completion of 12th (HSC).
https://www.shiksha.com/science/articles/what-is-bsc-course-all-about-know-the-full-form-eligibility-scope-blogId-13262

12th will be a good idea but the rest is your choice.

No, only your B.Tech result will matter.

No, it doesn't matter in all institutions as most of them consider the scores of graduation (B.Tech) but it matters in many institutes.

You can get admission in BA and after BA you can appear in many tests. If you are interested in technical areas, go for polytechnic courses.

Hello Namira,
To take admission in genetic engineering at graduate level you need to clear 10+2 with biology, chemistry and mathematics as well as genetics as part of the biology.
Most universities do not offer it as a special discipline but as a subsidiary in biotechnology, microbiology, biochemistry.