please confirm if I'm eligible for the 72 free transit

I will be staying in Beijing for 74 hours (11:15am Sunday to 1:30pm Wednesday), during which time I will travel to mutianyu to see the Great wall (is this still considered Beijing?) then travelling to Hong Kong. Will my family and I be eligible? We will be staying at a hotel so if we are eligible does that mean we don't have to register at the police station?