Will suggest you following points.
1. Take Gatimaan Express if you are comfortable in taking train from Nizamuddin Station. Gatimaan and Shatabdi Coaches are Same. With Gatimaan Having 10-12kmph more speed then Shatabdi and without any Halt. Food is served at Seat itself in both the train.. You won't find All seat around the Table. Only Middle seats have Table for rest There's provision for Seat Attached Food Rack.
2. Yes Taj Mahal is a Must , as i'm from Agra Only, Private Msg me i can give you some insight and help...
more... you regarding planning the trip.
you can visit Taj Mahal , Agra Fort + Sikandra Also, in One Day itself.
Will suggest you that you should take Bhopal Shatabdi in the morning as it will save your time at Agra.