Simple: register your bunq account as a bank account in the Paypal Wallet. After this account has been verified via two deposits from Paypal (typically they deposit two small amounts of a few cents and you need to register this amount in the Paypal environment once you have received them) you will be able to use this account to settle payments you make via Paypal. Also, when there is a surplus balance on your Paypal account you will be able to return this amount to your registered bunq account.

Since Paypal is not offering a true IBAN account it's not possible to transfer any money directly out of your Paypal account to a random IBAN account.

Exactly. PP originates from the US and is designed to pay other PP users world wide. They implemented an IBAN connection just for transferring money between your personal bank account and PP. You can also topup yourPP account via iDeal. The only way to transfer from PP to your personal account is as Arjan described.

How do you register your bunq account with Paypal? I've tried to use the IBAN sort code and account number but paypal is not able to find the sort code.

At bunq, we don't have a Sort code, as this is only used by banks in the UK 👌

Usually, changing the (billing) country for your PayPal account to the Netherlands will let you add your bunq IBAN correctly. If you still have troubles, you can of course contact PayPal too to check with them what's the best course of action in your case :)