Trivandrum

https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor?utm_source=connect&utm_medium=CR_JEECP&utm_campaign=4479470
You can use the above link to know about your future college based on JEE marks.

Yes. You will get admission in some good top self-financing engineering colleges in TVM.

Yes, Asian School of Business offers BBA course.

Admission is based on the marks obtained in graduation. Interested candidates can apply online with the required information.
Eligibility:
Candidate must be a graduate from a recognized university. Regards.

Please refer to the link provided below for details regarding the IISER Entrance Exam pattern, application process, question papers. https://www.shiksha.com/science/exams/iiser-entrance-exam.

Interested candidates can apply online with required information. (Website:- http://www.cet.ac.in/). Regards.

Please refer to the link below to know about the eligibility and other details:
https://www.shiksha.com/architecture-planning/course/b-arch-college-of-engineering-trivandrum-thiruvananthapuram-181429

You can just apply in any company. For getting details on job ads you can read the " employment news" news paper. It is very good and knowledgeable. It gives you all the details on jobs. This newspaper comes twice in the week only. It mainly give you the details on job which are offered by government sector. On internet you can go on Linkedin which is a very good site for searching more jobs by making your profile there. For any query please feel free to ask.