Mechanical Engineering

Bombay then Delhi Kanpur roorkee

You should opt for ECE.

Your category ?

It depends on where do you wish to work later. If you are looking for global exposure, master's from abroad will open all avenues for you otherwise Indian institutes are equally good as companies like TATA, Mahendra etc are training for rural market

To start with, take a look at some of the business schools in Germany. You will need resume, LOR from your employer, GMAT and TOEFL/IELTS score to apply.
http://find-mba.com/schools/europe/germany

Yes it is also a good college. It's placement is also quite fine. But remember one thing that college will give just a platform to show your skill not JOB. So focus on improving your qualities and try to express it so that you could be one of the best engineers. All the best for your future.