Is Max-Cut APX-complete on triangle-free graphs?
Yes, by a reduction from MaxCut to triangle-free MaxCut. Here is what Wikipedia calls an L-reduction

Given an instance $G$ of Max-Cut, construct the 3-stretch $G&#39;$ by subdividing each edge into three edges. Then the order of the maximum cut of $G&#39;$  is the order of the maximum cut of $G$ plus twice the number of edges in $G$. Since the size of a max cut is always at least half the number of edges, the error ratio only gets worse by a constant factor.