How do I extrude an edge that does not have an open side?

Point in case, how do you extrude an edge such as the one in the screenshot (the middle edge) downwards? The object is supposed to be a door frame, and I wish to make a curved door that fits the arch exactly. I was thinking of extruding the door shape from the door frame. Is there a better way of doing this?

And even if there is, is it possible to extrude an edge like this? I tried splitting the vertices, but this is the result

The new vertices are connected to the adjacent vertices, and I cannot reconnect them to the vertices directly above them. Is it possible to reconnect a vertex to a different vertex instead of the current ones? Can I remove an edge?

 

Delete one of the adjacent faces, extrude the side, then select opposing edges of the deleted face and Bridge Edges.

Door construction option :-
Duplicate Arch
Select inner faces corresponding to edge faces of door
Inverse Selection and Delete Faces
Select bottom two edges and Bridge Edges to complete edge faces
Flip Object Normals.
Fill Holes

But someone else will most likely have a better solution !

 

Thank you. It worked like a charm.