Does O(1) communication complexity imply that a language is regular?
No.  Let $L_0$ be a context-free language, say the language of matched parentheses, $L_1 = \Sigma^* \setminus L_0$, and
$$L = \{ij \mid b \in \{0,1\}, i \in L_b, j \in L_b, |i|=|j|\}.$$
Then $L$ is not regular, but it has communication complexity $O(1)$.