I have 15+ years of PHP/MySQL experience, including working with multiple frameworks and CMS, including some basic tasks with Joomla. I also have some experience in developing multilingual sites, and have created several booking/reservation sites, including accommodations. I have also integrated with PayPal on dozens of projects.
It sounds like you will need three new objects added to your database/controller, for the accommodation(s), bookings, and customers. I can create these along with the booking form, the payment form, and the email notifications both to your site booking administrator(s) and your customers.
There are some Joomla extensions for PayPal and Calendars, but I haven't found one that seems the perfect fit, so I don't believe using those would be better than creating a new self-contained accommodation extension/plugin.