Rereading your second post, this much can be proved VERY easily! The fact that tetration cycles around the fixed point can exactly be related to the exponential map.
Let
, if >0} \to \mathbb{D})
(where 
is the unit disk), It is periodic (first of all) and rotates around zero as we increase the imaginary argument, it tends to zero as  \to \infty)
, and is INJECTIVE modulo its period.
We get the exact same thing with tetration
, substituting 
and )
and 
. This is biholomorphic to the previous scenario via the Schroder function. It's no surprise that it behaves how you described. I always wondered what kind of crazy fractals it performed when it cycled around, but that much I couldn't plot. I just knew that many fractional iterations created a weird 
on some weird simply connected domain.
Let
We get the exact same thing with tetration