Vacation rentals system

I am currently working with a bought theme which was created for real estate websites.

This is the url:
http://demo.ehvoe-accommodation-chamonix.com/

The theme itself was originally designed as a rental accommodation (long term) and property sales.
I would like to turn the rental aspect of the site into vacation rentals, meaning that people should be able to search by the added criteria of date.

I’m guessing that the modification requires a booking system plugin.

1. there should be an availability calendar on rental property detail pages
2. A user should be able to search for accommodation with the added criteria of start and end dates.
3. After finding the right accommodation the user should see a form allowing them to send a booking enquiry
4. admin should be able to see the enquiries and change the availability status
5. it would be nice if there was the ability to use a payment gateway too.

Any help / advice would be greatly appreciated.