Computer Engineer

BE or B.Tech in computer science or IT is the required course to work as a software engineer. You need to apply for written test and then interview in various IT firms.

You can give any engineering entrance exam, but focus on JEE mains and JEE advance.

With B.Sc IT degree you have all each got basic graduation degree to be a software engineer. You can get entry level jobs in small and medium IT companies now. It is recommended that for a good career you may do either MCA or M.Sc IT/CS from a good college.

You should go for B.Tech/BE in computer science or information technology.

To become a computer engineer, you require PCM at +2 stage. Therefore you are not eligible to seek admission in B.Tech CSE. You may evaluate doing BCA. Some private universities allows non-maths students to do BCA.

IT is not a core function of majority of PSU. IT is only a facilitating technology to support core business of PSU. Therefore, in every PSU IT professional have limited growth and career opportunities. However, few government organisation such as Railtel, CDAC, NIC, NISG etc which are dedicated for IT, IT engineers have good growth opportunities.

There are plenty of government/PSUs jobs waiting for you. You must prepare yourself for different competitive exams to get them.
You can check the below mentioned links:
Computer engineer career options:
https://www.shiksha.com/career-as-computer-engineer-cc-31
List of government job available for computer science engineer:
https://www.shiksha.com/career-as-computer-engineer-cc-311
Best of luck.

Computer science engineering deals with the designing of software, looking at algorithms and how the two interface with other software. It is part mathematics, part engineering and a little bit of creative arts all mixed together.
Computer engineering is more of a hybrid role that focuses on how computer science and electrical engineering come together. Computer engineering deals with hardware design. The engineer focuses on both the hardware and software aspects of technology. They study how the two interact together.
According to me computer science engineering has more scope.

No, you cannot write engineer in front of your name.

BS means Bachelor of Science. It is same as BE. The terminology is different from country to country for different courses.