Never heard the one about men and women not riding together on the metro. There are some women only cars. But a man and woman can ride together in the other cars.

Use cleartrip.com to book train tickets. It works surprisingly well.

Yes, you can just leave your hotel and walk, but bear in mind that Indian cities are intense - hot, noisy, cars, buses, bicycles, rickshaws and in some cases, cows. Walking is not always relaxing. The touts may be outside tourist sites, but otherwise you are unlikely to be bothered much.

If you are a man and woman together I wouldn't be too concerned about safety.

You don't say how long you are going for, but if you have extra time, Udaipur is well worth a visit. Or if you are pressed for time, you might consider doing Agra as a day trip from Delhi (though many are sure to disagree.) Other that the Taj Mahal, it is a rather crappy city.