How did you make the picture? I don't know what information is in the x3d file I posted, since I don't know how to view it. But your picture looks very much like the original Maple graph I made the x3d file from!
Yes, I agree, the x, y, and z axes should be part of the model to help the students understand. I think a 1 inch size would be very useful. I envision having a collection of models of different surfaces and perhaps space curves. Perhaps I could have them all mounted on a board, with their axes all pointing in the same directions. But not permanently mounted, so the students could pick them up.
I assume from what you wrote that you can do 3d printing in colour! That is amazing. If it can't be done in colour, I was thinking of painting the models.
I don't think it is a problem that the surface has to have a thickness. It would still give students the right idea.
If the surface touched the coordinate axes in maybe only one place, would that be enough to keep the surface fastened in place? If not, perhaps I could make a few other gridlines. Is it possible to make parts of a model out of a clear material? I was thinking that coloured axes and some clear gridlines might make the model more stable.
Having clear gridlines might also allow the creation of a model showing points in space.
I was also thinking that sometimes it might be good to include the coordinate planes (in a clear substance) as part of a model, so the students could see how a surface intersects the coordinate planes.