Sending, receiving, and organizing email in Outlook.com
It sounds like this is most likely an Airbnb account issue rather than a Hotmail/Outlook issue. Microsoft doesn't have access to Airbnb accounts or the ability to sync or change the email address associated with them.
If you can successfully sign in to your Outlook.com (Hotmail) account and send and receive emails normally, then your Microsoft account is working as expected.
Since you originally signed in to Airbnb using Facebook, it's possible that your Airbnb account is still linked to your Facebook login or to a different email address than the one you're trying to use. I recommend contacting Airbnb Support again and asking them to locate your existing account using your full name, previous and current email addresses, old and new phone numbers, Facebook account, and any reservation or booking confirmation numbers.
If you're unable to access your Outlook account itself, you can try the Microsoft Account Recovery form: https://account.live.com/acsr
Otherwise, Airbnb Support is the appropriate team to help restore access to your existing account and upcoming bookings, as Microsoft cannot modify or recover third-party accounts linked to Outlook email addresses.