how much does it cost?

I be traveling with 7 friends. We have a day layover in china from new York. We wanted to take the day to go visit the great wall, we are staying in am apartment. So I have 3 questions. One with the 72 hours visa are we allow to do it? We be leaving again from Beijing to Singapore our final destination, so I wanted to know the cost of the 72 hour transit? And last if we are not staying in a hotel and only there for less then 24 hours do we still need to register at the police station? Thank you, Loreto.