The other answers are basically right, but don't directly answer the two questions posted by OP. I will try to do so.
(1) Why is Tt incorporated into the FBD?
A free body diagram illustrates the forces acting on a particular object. In this case, the object is the midpoint of the rope. This free body diagram must include Tt, because the top part of the rope is acting on the midpoint of the rope. (what else could possibly be holding the midpoint of the rope up?)
(2) Why does Tm point downward?
Since the free body diagram must include all forces acting on the object (in this case the midpoint of the rope), it must also include the tension on the bottom half of the rope. This part of the rope is pulling the midpoint downwards (because that's what tension does - it pulls in the direction of the rope).
Now more to the point - the FBD here is wrong. $mg$ should not be acting on the midpoint of the rope, unless we are claiming this point is somehow large enough to have it's own mass (more to the point, which mass is the mass $m$?).
The FBD of the 6 kg mass is correct, and the $m$ there is 6.0 kg.
The FBD of the 5 kg mass is correct, and the $m$ there is 5.0 kg.
From there, the correct way to solve this problem is to break up the mass rope into little sections, and determine the tension as a function of height $T(x)$, and then evaluate it at the halfway point. The answer is the same, but the concept is correct and more consistent with Newton's laws.