Hello.
1) In the Booking Calendar plugin, you can insert into the post or page, only the availability calendar instead of booking form with calendar.
Please read more how easily insert booking form or availability calendar into a pages https://wpbookingcalendar.com/help/inserting-booking-form/ or how manually configure Booking Calendar shortcodes in content of your pages: https://wpbookingcalendar.com/help/booking-calendar-shortcodes/
2) Relative syncing data from airbnb. Booking Calendar plugin is supporting
Please read more about this feature here: https://wpbookingcalendar.com/faq/#sync
Please note, the import process is working in that case, when some visitor is open your website page with booking import shortcode. So you can configure CRON at your server, for periodically access this page with booking import shortcode to start import process. Or you can insert the import shortcode before booking form shortcode, so firstly system will start import process and only then show booking form, when someone visit your page.
Export. Booking Calendar is generate .ics feed with bookings in real time. But how often some service (like airbnb, or booking.com ) access this .ics feed to import the bookings from Booking Calendar into own system, does not depend from Booking Calendar. So in other words, Booking Calendar does not force to generate import process by third party service, when some booking is created. You need to check with support of this service (like airbnb or booking.com), about how often they can access the .ics feed tor making import.
Kind Regards.