The cloner on the right does not have any effectors in its list.
Drag the effectors you want in there manually.
This happens when you don't have a cloner selected when you add the effector, or add a new cloner after adding effectors to the previous one.
CBR
Yes, Disconnect would also do it, but I remain slightly confused about why. If we look at the problem mesh on the right there, there are actually borders along all the edges of the cube, so I am left wondering why Cloth was doing what it did there. Oh well, I'll put it down to quantum weirdness, and let it pass...
CBR
They are merely disconnected poly islands within the same mesh. And they really have to be disconnected for this sort of panelling to be useful / work.
No, it shows the opposite of that, the green lines being being borders ie places where polygons are NOT connected to the adjacent ones.
Yes of course; seletcion tools in component modes don't care what is connected to what - you can select it regardless.
The answer to getting the result you wanted above, ie preventing those cross-corner parts is to split that mesh into either 3 or 6 separate meshes. So either pairs of cube faces on each axis, or each cube face individually. That way they can all go under a cloth SDS, and it will be impossible for it to add thickness across corners.
CBR