Hello.
Please note, Booking Calendar is use standard WordPress wp_mail function for sending emails. This function can be overridden by other plugins.
Please check this troubleshooting instruction https://wpbookingcalendar.com/faq/no-emails/
The most probably you need to check the point #5 from this instruction and install WP Mail SMTP plugin (or alternative), which is ‘sending emails via php’ option and it can resolve that issue.
Kind Regards.
Hello.
You just need to add widget for «availability calendar» only (instead of booking form with calendar). In the WordPress > Appearance > Widgets page in «Booking Calendar» widget for the option «Show» set «Only availability calendar» instead of «booking form with calendar».
Or if you have inserted the Booking Calendar shortcode (and do not use widgets), then you just need to insert other shortcode: http://wpbookingcalendar.com/help/booking-calendar-shortcodes/#bookingcalendar
Kind Regards.