I created all the 3 faces attached to the extruded cube with shift+F. I tested some variations and the results were prettym uch the same. It uses some edges to make a nice smooth subdivision and it doesn't use others as subdivision orientation. There, it's sharp. Why is that? How can I make it round where it's sharp? This is with faces inside. This time, it worked somehow. There's still a lot of sharpness: When the inside faces are removed, it spikes again. There are never zero value faces (shown):
Your biggest problem are these internal faces. I can't really think of a reason you would ever want something like this, so start by deleting them.
The remaining problem, faces have fronts and backs, and the reason you're getting that cusp is because some of your faces are pointing the wrong way.
Select the whole mesh with 'A' and hit 'Shift N' to automatically correct the surface normals.