VLSI Design

You can choose VISIT design in Osmania University. Good luck.

Yes, you are eligible to apply for this exam. If you are interested to pursue your career in research or teaching then apply for the UGC NET Exam. The minimum eligibility criteria to apply for this exam is post-graduation. The exam is conducted twice a year (June and December). You can apply for both these session in you want.

You can use college predictor to know about this:
https://www.shiksha.com/college-predictor
Good luck.

I don't know if I am qualified to answer this, but as a student in the Electronics field I have some idea about it. If you want to go for VLSI, see the places where there's good openings for VLSI after you graduate and the ones who are doing good research in the field. The two best places in my opinion to read VLSI will be US and Taiwan. Because along with good research they have plenty of semiconductor companies who are involved in the work.