Thank you,
I think this also answers the first question. It will not work when there are discontinuous wall contacts.
What I meant with a "gap" was fluid between the tip of the tool and the interface. Without this gap the interface is made of wall surfaces (tool) and fluid surfaces (fluid between teeth). With the gap there is at least one layer of fluid cells to avoid direct contact between the wall of the workpiece and the wall of the tool.

I tried using the overset mesh approach previously, but the same wall-wall contact generated orphan cells, which caused the solution to diverge. I think this would be another discussion.