The most common technology used to integrating with 3rd party applications is by Web Services.
Since the booking process is the core part of the hotel application, you would have them expose a secure web service with operations that allow you to perform a booking.
Example of these operations would be:
get all suites (retrieves a list of suites)
get availability (this allows you to identify the availability of a suite)
reserve suite (reserves a suite for customers, payment is unconfirmed)
confirm reservation (confirms that payment has been made)
Yes payment can be made via a payment provider; the provider would have a payment service that you should be able to use. The travel app is only acting as an intermediary, if commission to travel app is applicable you should negotiate the terms with the hotel and establish how commission would be payed back to travel app; either after the fact or whether the payment should be made to the travel app and then the travel app would pay the hotel.
design patterns involved
This highly depends on your interaction with other systems.
Things to think about: make sure that the customer has been authenticated, you don't want bookings to be made and customer not paying for the suite, leaving the hotel fully reserved with no one staying at the hotel.