The logging wood crane is just an animated object with no particular asset associated with it. There are wood lot assets from various content creators that you can put around to simulate the crane picking up the logs and placing them elsewhere, otherwise, as you've noticed the logs vaporize into thin air.
Regarding log spine cars:
It depends upon which era and what kind of railroad you are modeling. I know there are some narrow gauge spine cars available on the DLS. You can search for them directly, or better yet, download the Clear Lake Logging model railroad route by Phil Skene. (philskene). This will give you the spine cars and other equipment and show you how to set up things including the logging crane.
For more modern routes, I'm not sure what's available on the DLS, but if you are interested in DLC, or perhaps directly from Jointed Rail, check out the Legend of the BNSF 2, and Eagle River. This route has the modern spine cars and some bulk flats. Both of these assets accept logs and wood and work interactively with the industries.
