I made the page in English initially, then installed WPML, and created a Dutch translation. When I now switch the default language of frontend/backend to Dutch, the search stopped working and is not even showing on some pages, and the accomodations are not available on the calendar. Is it possible to switch the default language like this? Or do I need to start over?
You should make sure to add translations for Accommodation Types and Rates, otherwise, there will be no availability in Dutch language when switching to it from English. Check this article with detailed information on how to translate everything needed via WPML.