Pagalguy (https://www.pagalguy.com/discussions/mba-in-india-16898143)
I may not be able to answer both your questions directly, but surely could let you know one thing that you have ample opportunities of working with startups in any B-school if you want to take those opportunities up. The number of startups visiting bigger and more famous b-schools is less as most students want to go for established companies. (Most, not all. Many aspire like you too) However, opportunities for doing your summers and live projects is always open wherein you can actually learn by working with startups and then decide for yourself if they suit you well. This will automatically direct you to the startups coming for placements and give you an edge.
Since I am a marketing guy, I think marketing is somewhere you can grow manifold and make a good career ahead, a tad bit more than other streams. Operations seem to be fine as well. Having said that, people do wonders with streams like HR and IT too, so there’s no end to possibilities. E-commerce, especially in tier 2 colleges, has a great presence in placement books, so you do not have to worry about that before taking admission in any of the known b-schools.
Hope this solves at least some of your purpose(s).