Tiled 0.16.0 not save custom properties?

This is intensional. There is no point in saving properties with their default values, since this would result in needlessly verbose files. Imagine all your objects would have all their default properties stored, when you have many possible properties and many objects on your map.

The idea is that generally the properties and their default values are already implicitly present in your engine. If not, then you could still just read the object types XML file stored by Tiled in your game to get these values. For this you would change the location of the object types XML file to somewhere inside your project directory.

Btw, in your particular case you’re using a custom property to refer to an image. Instead, why not add this image to an Image Collection tileset and place it as a tile object?