With regards to how long of your time you will spend on the Taj Mahal, there are two options. One is the half day tour which is from 10:30 AM - 3 PM. And the whole day tour which is from 10:30 AM - 6:30 PM. The half day tour only tours the Fatehpu Sikri whereas the whole day tour covers the Taj, Agra Fort and Fatehpu Sikri.
Picture taking is allowed on monuments. Taking videos is fine but it there is a fee for the video camera.