1. Three days for Delhi and the Taj Mahal is just fine.

2. Any airline should be fine, besides Kingfisher (they are close to bankruptcy and frequently cancel/delay flights these days). There are no *A carriers in India, but certain classes in Jet Airways will get you UA miles (not EQM).
A good place to search for Indian domestic tickets is makemytrip.com (make sure you are on the Indian version of the site since this will have better prices).

3. Not much in Chennai. I would recommend an extra day to see Kanchipuram (famous south Indian temple and very different from what you will see in/near Delhi) and Mahabalipuram (beautiful, famous, ancient temple+stone carvings). You can easily cover these two places in a day trip from Chennai.