Why is it that when I extrude individual faces, I am getting border with a discontinued profile at the corners (see picture)?
Is there any way to make the border continuous with uniform thickness around a given object "P".
If you extrude individual faces these faces will not be connected anymore. This is why you get discontinuities.
Extrude your faces by pressing E instead (this is equal to
Extrude and Move on Normal). Then immediatly press Rmb or Esc to cancel the translation. now the extruded faces should be at the same location as the previous faces.
To get walls with an even thickness use
Shrink/Fatten by pressing AltS. By moving your mouse you can change the thickness of your walls. To get even thickness independent of corner angles hold down the Alt key while moving the mouse. If you are satisfied press Lmb to apply the tranformation.