I would like to extract out the icon in the following screen (mixed with gray background)
Anyone know how I can do that using GIMP?
Personally, I would rebuild the image from scratch because I find it frustrating not to have original source files. But, until you or someone else gets around to doing that, here's the non-destructive quick fix:
Unfortunately it is never perfect to extract elements out of a raster image. Others will hopefully chime in with the best way to accomplish it in GIMP, but do note that for various reasons (especially anti-aliasing, which smooths out lines by blending pixels on the edge of the line with the background) any attempt to select part of a raster image and delete the rest will always result in either a ragged stair-stepped edge or an ugly halo of the background color.
In this case I would use a layer mask to remove parts of the image. The free select tool with polygon snap (polygonal lasso tool in Photoshop) can be used to paint black/carve off straight sections from the right side of the image, and then use the fuzzy select tool (magic wand in Photoshop) to get around the tricky gaps in the top-left. Not perfect though.
For this reason you always want to keep the original image with either intact layers for later modification or as a vector illustration.