So I just want to fill that path with color but this happens. When it is not filled or in 0% opacity, it looks really normal, no doubled edges or things. I don't know how to fix that.
I opened your pdf file in Illustrator.
Select your image and go to Object> Compound Path> Release
Then open your layers panel and you will see 3 separate Paths
Each of those 3 Paths have 3 Transform Effects applied and this is causing your trouble. You can see these in the Appearance Panel. I toggled off the visibility for 2 of the paths in the Layers Panel and selected the other visible Path. Open the Appearance Panel and toggle off the visibility for the 3 Transform effects that have been applied- I would simply delete those Transform Effects for each of your 3 paths.
Then your Fill will precisely follow your path outlines.