I have this same issue, weird thing, at work i can open the file just fine and it shows transparency At home i get some green hue and its opaque??? Im running crazy... same file, same photoshop versions, both open it differently?!?! Preview on OSX handles file just fine, thats even weirder. SO i copy pasted between Preview and Photoshop to get it looking ...
I'd try the following approach:
invert the selection, to make sure only the head is selected (currently anything but the head is selected)
paste as new image
From there, you can export it as usual, if the image turned put to be as expected.
GIMP does currently have no way to directly export selected parts to image files, unless you want to venture ...
You can't remove that in Illustrator, because Illustrator is built around editing vector files, and PNG is not vector artwork, it is a raster format (made of pixels).
You can however, attempt to do this in Photoshop — which works with pixels — using the 'Levels' panel.
The way I do it:
Inside Photoshop, make a selection and make a path from selection.
Copy that path and paste it in Illustrator.
Use the script GetShapeArea to calculate the area.
Note: In the image below I altered the code to show just cm². But the code has in² and cm².