I would like to create one ZP album in which I can drop all kinds of images, just to use them in a zenpage, but I don't want that album to be visible or accessible for visitors. Is that possible by just unpublishing an album? Will I still be able to access the images in the way I would like to?

Your best option is to both un-publish and password protect that album. Unpublished only hides the album thumbnail. Visitors who know the name can type the link directly and view the album.

To answer the question directly: Yes, you should be able to use an unpublished "hidden" album for that purpose. Or you drop those images into the uploaded folder and use the file manager (see the screenshots section).