How to integrate a booking form? ... preferably with PHP. I am new to Drupal.

My goal is to integrate a page into an already existing project which enables to give to users the possibility to book a Hotelroom or a flight ticket while the Administrator (me) then collects the submitted information and validates the choice in the backend.

I have done this countless of times from scratch in PHP with POST. But on Drupal, when I try to integrate HTML and PHP code into a new webform or new article page, it doesnt really render it.
After reading your question I would do the following:

Create a custom module with a custom block that holds the booking form. This way you can add the booking form on whatever page you like with the Context module for example.

The Drupal 7 example module can show and teach you how to create a block in a custom module.