Hi,
This problem isn't specific to the toolbox, but just trying to use svg in html. I'm not an expert in svg, so maybe someone who knows more can offer more insight, but from what I can tell, trying to use css on an external svg isn't supported. You're using an external svg because you're referencing an external URL. However, I don't think your markup is correct. You can either use the img tag, like you did get working, or an object. For example, the following code works for me:
createIconDom_() {
var toolboxIcon = document.createElement('object');
toolboxIcon.setAttribute('type', 'image/svg+xml');
return toolboxIcon;
}
Otherwise, you could inline your svg, so instead of having an <object> that references a file, you put the svg code directly in. If you open your svg in a text editor, you'll see all the svg code. If you were to put that directly into the code, it would be an "inline" svg, and you might have an easier time styling it.
https://css-tricks.com/using-svg/#using-inline-svg
So to summarize, this problem is related to styling svgs in general, and not a problem specific to the toolbox. The toolbox should support an svg in the icon dom.