Two more hopefully helpful questions.

Should have been the first Question I asked but I overlooked it. When you select that Object/Face and then hit Extrude do you get any message in command line down the bottom? You might have it turned off (from yhour screenie) Display>UI Elements>Command Line to turn it back on again.

If theres a reason its not doing the extrude then it will say why there, and that will also explain no handles.

My other question is if it IS extruding then your verts should be doubling every time you've hit the button. if you go Display>Heads Up Display>Display Polygon Count You can select your object and it will tell you what its made up of.. This probably wont help you, but it might help me...

K PROMISE I'm shutting up here now and going to bother David or Nilla