You can see the path if you do Edit > Preferences > Tools > Node, and check the "Always show outline" option.
You can't easily have a path with an invisible stroke on one segment. A path only has one stroke attribute. I mean it would be possible with tricks like clipping paths or masks but you probably don't want to go down that route just ...
Let's assume your description of your situation is valid. Then your shape should be a single path (=a Bezier curve) which is filled with orange and has a black stroke. The black edge shouldn't be a rendered effect, it should be seen in the stroke and fill panel like in the next image (the nodes are visible because the node editing tool is active)
In the ...
In Illustrator add anchor points to all elements a couple of times.
In the menu bar go to Object -> Path -> Add Anchors
Then, in the SVG options panel, increase the Decimal Places to the max (7 I think?) and see if the results are any better.
In Inkscape, you can select both paths, and do Path > Stroke to path.
This will convert the strokes to outlines, with a fill.
Next, do Path > Union. This is a boolean operation which will merge the two paths to one solid shape - i.e. a single closed path with a fill.
Now you can apply a coloured fill, and a thin black stroke to get the effect you ...
ImageMagick's convert function is going to call Inkscape for SVG images, which doesn't know anything about relative unit measure (such as ex). Nonetheless, browsers are very good with those units. Meets svgexport, a Node.js module and command-line utility which is perfect for this job. I converted a few hundreds images in a matter of minutes.
The extra space is the extent of the gaussian blur filter that has been added to the highlight in the middle of the crowbar.
To be honest, you can just ignore this. It shouldn't stop you from just rotating the image slightly. Alternatively, if you don't want that gaussian blur filter, select the object, and just delete the filter.
I think part of the problem here is that the SVG wasn't generated in Inkscape. Messing around with other people's or other applications' SVGs in Inkscape can get really messy sometimes.
Personally, I'd just recreate it in Inkscape since it's a very simple design. It would certainly be easier than trying to edit it. Then save using the Optimised SVG option in ...