004 is about the thickness of a sheet of typing paper. You can shim it with the paper to figure out some of the other issues prior to getting anything hot enough to solder.
I figured .012 was a mistake after seeing the photos-I don't think the dolls head would fit into the frame at .012 wear.
I'd likely disable the ejectors until I figured out what I had and what it needed. There are few times an ejector is really needed. Never, at my house.
You can approach these projects from the heart (usually ending up with more into it than it's worth) or, from the head, with just enough into the gun for satisfactory performance.

Best,
Ted