I figured it out only after I zoomed in your image, the border you are seeing isn't actually a border, it's a rasterization error found in Illustrator and there isn't much you can do about it. (see my answer on that topic here).
Also, setting the opacity of an object to 0 won't make it disappear
and when you can still select it and see it's outline (the blue one).
In your case, trying to figure out an easy way for you to get the desired result is to:
- Make all your shapes 100% opaque again and select all of them
- Go to Pathfinder > Divide
- Ungroup the resulting shape and delete all the other shapes not needed for your desired design
- After you got your moustache design, select all the remaining shapes and go back to Pathfinder > Unite to unite all of them back into a single object.
A very fast example of something looking like yours so you get my point:
This might not be the cleanest way of doing it or the fastest but it will get you there with little trouble since your design isn't that complicated.