Non-monotone evasive graph properties. Are there examples of non monotone graph properties which are evasive?
Since AKR conjecture doesn't mention them, there should be.
There is this property: "Is the graph a spanning tree?"
Is this property evasive? Clearly it is not monotone. But it seems to be evasive since spanning tree is the output which adversary gives in case of  "Is the graph connected?" These two properties are thus related.
Is there a good example which is not related to some monotone property like in above case?
Is regularity evasive?
 A: The property "has an even number of edges" is non-monotonic and evasive.
EDIT: 
While the asker's original question was about non-monotone graph properties, it appears his interest has further extended to exclude properties which are unions of intersections of monotone graph properties. Unfortunately for the asker, every graph property can be formulated as the union of the intersection of monotone graph properties.
Let $P$ be a graph property (considered as a set of graphs). Let $P = \cup_k P_k$ where $P_k$ is the set of graphs in $P$ containing exactly $k$ edges. Let $A_k$ denote the set of graphs $G$ with $\leq k$ edges such that $G$ can be extended to a graph $H \in P_k$ via the addition of zero or more edges. Similarly, let $B_k$ be the set of graphs $G$ with $\geq k$ edges such that a graph $H \in P_k$ can be obtained from $G$ via the deletion of zero or more edges in $G$. By construction, $A_k$ and $B_k$ are monotone and $P_k = A_k \cap B_k$. Hence $P$ is the union of the intersection of monotone properties. 
