Axial deformations are ignored. This structure is under the specified force:
For your first question regarding whether the shear in BC equals the shear in DE: yes, they are the same.
An easy way of seeing this is that some fraction of $P$ is going to be absorbed by BC as shear, which will then be transferred to CD as an axial load, only to then become shear in DE again.
As for your main question: you haven't included all the loads acting on that slice of the frame. You've drawn the moments, but forgot about all the other forces acting on that part of the structure. Specifically, the axial and shear forces at the cut points (mid-BC and mid-DE).
If you were to solve this entire structure, you'd find that BC and DE are under axial loads due to the support reactions (ABC under compression, DE under tension). These opposing loads are a force couple which generates a moment equal and opposite to the moment at mid-BC:
In the example above, we have that mid-BC has a bending moment of $\dfrac{133.2}{2} = 66.65\text{ kNm}$. However, BC and DE are under 13.33 kN compression/tension, a force couple which generates a moment of $13.33 \cdot 5 = -66.65\text{ kNm}$ (negative because it's clockwise), perfectly cancelling out the bending moment at mid-BC.
Diagrams obtained with Ftool, a free 2D frame analysis program
