Computers & IT (Non-Engg)

You should apply for National colleges like:
St.' Paul's College
Wilson College.

There is not much of a difference in BCA and engineering. B.Sc has a different syllabus. If you have any interest in pursuing engineering then go for it. Else, B.Sc will be a better option for you.

No, mathematics is not compulsory. You can also take PCB and do B.Sc. B.Sc in maths requires a thorough knowledge of mathematical concepts, but other courses or streams do not.

Diploma in Software Engineering or Computer Science or Information Technology is not a sufficient qualification for seeking jobs even with CEH certification and to become an ethical hacker. You are advised to take Science with Maths stream and then do B.Tech CS/ IT or BA-MA or B.Sc-M.Sc in CS/IT before seeking CEH certification. Path suggested may look long and hard but it will be helpful in the long successful professional career.

You should do ethical Hacking from any private institution, because private institutions can give you placement or you can opt for ANDROID or ASP.NET. They are also high earning short term courses.