Hi,
Im trying to extrude the inside of a shirt i modelled, but everytime i extrude a face, the source face disappears. I could fill the created holes manually, but it is a pain in the ***. Here are some picks to show you what I mean:
instead of manually extruding the faces, try using the solidfy modifier. it automatically does that and it allows you to easily adjust you mesh.
good luck 
That’s how extrude is supposed to work. Otherwise you would end up with edges connecting to 3 faces, which is not good.
Why would i end up with edges connected to 3 faces? Normally, when I extrude a plane i get a cube - and not capless box. Thats why i am a little confused by this.
because in a plane, each edge is only connected to one face to begin with. In your case the edges are connected to two faces, and if you would extrude and leave the original faces, the edges surrounding the extruded region would be connected to three faces.
I would also check you have removed duplicates and recaculated normals