For which k is PLANAR NAE k-SAT in P?
PLANAR NAE $k$-SAT is in P for all values of $k$ . 

Reason is that we can reduce PLANAR NAE $k$-SAT to PLANAR NAE $3$-SAT. Let $\phi$ be an instance of PLANAR NAE $k$-SAT, and suppose $\phi$ contains a clause $C$ with literals $\ell_1, \ell_2, \dots, \ell_k$. Introduce a new variable $v_C$, and replace $C$ with two NAE clauses $C_1$ and $C_2$. $C_1$ contains $3$ literals $\ell_1$, $\ell_2$, and $v_C$, while $C_2$ contains $k-1$ literals $\bar{v}_C, \ell_3, \ell_4, \dots, \ell_k$. It's easy to see that $C$ is satisfiable iff $C_1 \wedge C_2$ is and that the transformation preserves planarity. Now, we can repeatedly apply this procedure on the clauses to eventually obtain an instance $\phi&#39;$ of NAE $3$-SAT as desired.