I just recognized that the upper radio buttons do not send data either. So it's radio buttons and checkboxes that do not work. The input fields (and even the acceptance checkboxes at the end of the form) work properly.
You need to know that we had WP Super Cache running for a day. I stopped and deactivated it since I thought it might be the reason for the CF7 problems. I also cleared the cache before deactivating. Didn't change a thing.