Build booking system from scratch

Hello, I need to build a booking system equivalent to: [login to view URL] There must be 5 different types of prices (in the link there are 3), each price can be different depending on the day and time selected by the user. There are common availabilities for all people. Let's say that for 2pm today there are 20 availabilities: if we have a booking of 2 adults + 3 kids, it will deduct 5 from total availabilities. For each type of price, there must be an option to confirm if the selected number is deducted from availabilities (there can be 2 babies for example, but i don't want them to be deducted from total availabilities). Time slots must be customizable for each day. For example, today we could have 2pm and 6pm, and tomorrow we could have 10am, 11am, 14am, 18pm. The length/duration of the timeslot is not important. Every time the user changes the number of people, the system checks again availabilities with database. Same if user changes the date. When the user has made the selection and if there are availabilities, the "add to cart" button appears. When clicking on the button "add to cart", selection will be added to a wordpress cart (if feasible) as the main website will be on wordpress. When added to the cart, the number of persons will be deducted from total availabilities during 20 minutes. If after 20 minutes the payment is not made, cart empties and tickets are released. If payment is made, number of tickets purchased is deducted from total availabilities. Back end will have to be clear and easy to use. There will, in addition of the configuration of the days/timeslots, a reservations summary.