Consider the rigid relation ($RR$) principle, i.e.

"every set admits a rigid binary relation", that is,"that for every set $A$ there is a binary relation $R$ on $A$ such that the structure $(A,R)$ is rigid, meaning that it has no nontrivial automorphisms" ("that is, no bijection function $\pi$: $A$$\rightarrow$$A$ such that $aRb$ iff $\pi$$(a)$$R$$\pi$$(b)$, other than the identity function"). (Hamkins and Palumbo, arXiv: 1106.4635v1 [mathLO]).

In that paper, Hamkins and Palumbo prove that $AC$$\Rightarrow$$RR$, and that $ZF+{\lnot}AC+RR$ and $ZF+{\lnot}AC+{\lnot}RR$ are relatively consistent with $ZF$.

Question 1: What role (if any) does $RR$ play in the proof of the Kunen inconsistency?

Question 2: Since $ZF+{\lnot}RR$ is relatively consistent with $ZF$, is $NGB+{\lnot}AC+{\lnot}RR$ ($RR$ appropriately defined for $NGB$--call it $RR_{NGB}$) relatively consistent with $NGB$?

Question 3: Can the Kunen inconsistency be proved in $NBG+{\lnot}AC+{\lnot}RR_{NGB}$?