After going through your post I will say it is very difficult to visit India in one week. But you can visit some part of India. In one week North India region is suitable for you. you can easily spend time in the states like Rajasthan, UP, Himanchal, etc, but before that you need to book flight according to your time.

It is also difficult to travel without help of tour operator. so you need to take suggestion from a travel agent and they can help you.