NSE has more liquidity and quality of companies listed is generally better (not talking about the blue chips or good mid/small caps). It's easier to get listed on BSE compared to NSE since NSE has a more stringent criteria for getting listed.

If you're interested in a specific stock which isn't listed on NSE then you anyway don't have a choice, however as a rule of thumb, I trade on NSE. NSE/NIFTY 500 is a good universe to pick stocks from if you're into trading (intraday or short term) since volumes are good and you don't need to worry too much about operators controlled stocks. You can additionally use daily average volume as a criteria to filter liquid stocks within NIFTY 500.

Hope this helps.

There are plenty. Every broker has either their own charting kit of integration with another popular charting software. Zerodha has kite (chartiq) for charting and also have an integration with TradingView. 

My favorite is TradingView. It has a free plan and NSE/BSE data is also available for free. Plus, it's web based so works everywhere. Last but not least, it has programmimg (pine) support for backtesting and custom coding your indicators and strategies. You can have your personalized setup and everything is saved in your account so you can access it and trade from anywhere. It also lets you paper trade.

Other popular charting software include Amibroker and Metastock. In terms of features, they probably have everything you would even need, including F&O, option chains, OI analysis, backtesting and so much more. Amibroker is arguably the better option but the issue is that they work only on Windows. If you're on Mac or use a linux flaour (e.g. Ubuntu) then it won't work.