I don't remember if Cave Moss ever did grow down naturally like Hangers or not, but I guess I was at least under the impression that it did when I originally made its wiki page. I think it would make sense either way, so that cave moss is renewable.

True, making the naturally generates ones not grow would be good

They never grew, but it would make sense to add that

I would want naturally generated moss to not grow though (through use of a new blockstate property which would be set at the bottom block to not grow) otherwise caves would get messy.

Well regular Moss and Hangers already do that, but Moss also dies out, so maybe Cave Moss could have a chance to die out as well. Although Cave Moss doesn't seem to spawn in very big clusters so I don't know if it really would get that messy.