But I have no idea on how to make this part. What I want is to extrude some sort of flat 3d object out of the background image:
Create two curves, the baseline and the profile.
Create a bezier circle, disable cyclic, notice how it changes the shape due to having auto handles, so after rotating it to be symmetric on $y$ axis, switch the handle types to FREE and only then delete the two excess vertices.
Just duplicate the object and scale to $0$ on $y$ axis (will translate to 0 height on $z$ axis), then also scale it a little on $x$ axis (can adjust later). Notice how I switched the pivot to 3D cursor, to make the scaling also move it towards the origin (use .D).
Now set the Bevel to this profile:
Subdivide (F9 to adjust the number of cuts), scale, move, adjust curve radius (⎇ AltS or numbers panel) to your (p)reference:
First of all, if you have a clean image of an icon with flat colors like this you can convert it into a .svg and import it into Blender as a curve and/or convert the curve to a mesh. You can convert the image to an svg with inkscape, which is a free program. Here's a tutorial: https://www.youtube.com/watch?v=10m_2bPXa1s
Otherwise, if you want to trace this kind of detail there's many way to do it but I'd use a curve.