Ok, so you do want it to etch only the outline. To do that, don't do the Stroke to Path step.
After you do Trace Bitmap, that's basically the path you want, although you may need to tweak it a bit, here and there. (Select the black object, add a stroke and remove the fill, to see just the path. Object menu > Fill and Stroke, and also Help menu > Inkscape manual, with an internet connection, if it's not obvious how to use the dialog.)
You can tweak it with the Node tool, if necessary. Notice how you can drag the nodes around and that certain nodes have handles, which can be adjusted as well, to change the curve. If you haven't tried that yet, here's a tutorial I wrote as an introduction to paths. It's quite thorough and long, but covers almost everything.
http://forum.inkscapecommunity.com/index.php?action=articles;sa=view;article=7I still don't have a clue about what the problem might be, at the moment. I think you probably have paths that should work. Maybe without the double paths, it will work better?
I've heard about one particular type of cutter/plotter that doesn't like a lot of nodes. Fyi, I'm just learning about this way of using Inkscape, as I just took over as admin on the Inkscape for Cutting Design site, and there is a LOT to learn. Some of them will only cut paths of a certain color or width. Another requires the paths to be enclosed in a rectangle. It doesn't cut the rectangle, but if it's not there, it won't cut. There seems to be all kinds of little peculiarities for these machines and their software.
Anyway, let me know if there's any change, when you don't have the double paths.
Wow, so with a line being etched so thin, how do you accomplish a wider etch line? Do you have to draw a series of paths that are incrementally larger/smaller? Or can you just tell the program to make several passes at certain intervals? Sorry, I'm just so fascinated by these toys -- I have a million questions.
Oh, I did notice something else in that file! You have arrows pointing to the page border, indicating coordinates for 3 points. On the y axis (0,100,0) and the origin (0,0,0). But for the x axis, you have (100,0,-1.5). I'm not sure what the purpose for indicating those points is. Is the -1.5 the depth for the laser to cut?
If so, why is it only at that point? Could that be the problem?
A couple more things I noticed, but they're probably nit-picking. The document units are cm, but those coordinates are mm. And also, whatever those points are, they don't seem to relate to the drawing, that I can see, at all. They're probably just for reference, I guess.