You are not an NRI. NRI means non resident Indian. So NRI rules won’t apply for you.
NSE/BSE are just 2 different exchanges, with many common stocks and ETFs listed on both. So you can buy on one and sell on another. NSE is bigger than BSE so has more people ready to buy, sell, etc. More liquidity
Zerodha allows you to do SIP into an ETF or a stock. But not sure whether Zerodha can be used by US citizens.