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I created lines using the line segment tool in Adobe Illustrator (latest Adobe Cloud version) and I can't change the color of the stroke (through the Appearance panel) if the stroke weight is less than 2.0pt.
If the stroke is 2pt or more, then the color around the line changes but not the line itself.
The whole behavior is odd for me as I've never faced in Illustrator before.
Edit: added screen below with the line selected. 
In both screenshots you have deselected the line, so we don't see the appearance of the specific line.
It sounds like you have by accident added two strokes to the line. A black 2 pt stroke and a green stroke which becomes visible when it's larger than the black stroke:
The reason you can not see the line color applied to a thin path is because you have an appearance attribute applied to the whole layer and this affects all the elements.
To remove the layer attributes, click on the circle to the right of the layer name (target icon) and from the Appearance panel click on the forbidden icon (Clear appearance icon)
The Target icon, the little round icon displayed at the far right of each individual layer in the Layers panel, is used to select every item on the layer. Click the circular Target icon, then apply a stroke, change a color, or apply a style to all the objects on the layer.
