B.Tech

Yes, you eligible to do B.Tech from IIT-BHU but you have to clear its eligibility criteria. So go for this.
All the best for future.

Sorry to say but in India, there is no such engineering course. You can have certification course from some academies and institute in Delhi, Pune, Mumbai, Bangalore. You can also go for Diploma of 1 year. But B.Tech is not there in India for sound engineering.

You can pursue automobile engineering. The eligibility of the same is as follows:
The students, who wish to pursue automobile engineering, should have qualified class 12th with minimum 50% marks in Physics, Chemistry and Mathematics from a recognised board in India. Candidates, who wish to join IITs, should have cleared JEE Main and Advanced. Whereas, admission to NITs depends on candidate's JEE Mains score.

You have to complete 12th first with subjects maths, biology, etc. Then, you can.

I'll suggest you opt for any branch in a good IIT as there you can get your branch upgraded if you are among the top students of you branch plus the pack of top IITs is awesome whether it is core branch or not.

The eligibility is:
1) Candidates must clear the JEE Advanced exam.
2) General category candidates must score at least 20% marks in JEE Advanced i.e 73 (varies year to year) marks out of 366 and at least 6 marks out of 122 in each of the three subjects.
3) OBC-NCL candidates must score at least 65 marks out of 366 marks and 5 marks in each of the three subjects.
4) SC, ST and PD candidates must score at 10% marks i.e. 36 marks out of 366.

As per my knowledge and my opinion, to become a professor in Government University, you can go in any of the two options, you have mentioned, as if you go for B.Tech M.Tech and then Ph.D. You are eligible for professor in University of Technology, and if you go for B.Sc M.Sc and Ph.D, then you are eligible as the subject teacher of the respective subject in which you have completed the graduation post-graduation and Ph.D.

I also belong from the same field. So, I would suggest you to just be thorough with your core subjects, focus more on basics and do your internship programs carefully and you will be done by the time of MS. All the best.

You may attend the counselling conducted by the government on the basis of Andhra Pradesh PGECET ranks and opt your desired university.

After B.Tech 4th year, you can only get the starting salary of about INR. 4-5 LPA. As for the freshers this is also a huge amount in Trimax, you can easily get this pack, but you can search for some other companies also in the Pune or Delhi side.