When creating an new "empty page" and name it exactly like an image in that folder (except the file extension), the new page gets a thumbnail from the picture. This goes for folders also.
In the attached pic, in order: Image, a page, a folder all with the same name the last two is only for showing how the page and folder icons should look like

If I rename the imagefile the pagefile and folder loses the connection and gets the "normal" representive pic after I update, but not vice versa. This is only in jalbum desktop program, not in the published page.

It's extremely confusing for me. But maybe it's supposed to work this way?

Think about this for a moment. Let's say you have mydog.jpg, mydog.mp4, and mydog.pdf, all in the same folder. The core produces a thumbnail for the image: thumbs/mydog.jpg. Then it produces a thumbnail for the video: thumbs/mydog.jpg. Then it produces a thumbnail for the PDF: thumbs/mydog.jpg. See the problem?

As David suggests, the core should warn about this condition. I wouldn't want it to start guessing about how to fix it, however - the user might want to handle it differently in different situations.

But you run into similar problems with links to folders, extra pages, slide pages, and so on. There are all sorts of landmines lurking in there when you have duplicated base names of any type.

The core should be able to spot these, and warn the user. It gets tricky with the added pages, however, because the conflicts might not crop up until you get into a subfolder. If the skin is reusing a page that's been added to the root, the core might have difficulty predicting the clash.

For anyone else reading this thread see Naming of files and folders
for more information on avoiding problems with file naming.

jAlbum now warns about base file name collisions in v39.7. See https://jalbum.net/software/release-notes