The authors study holomorphic maps \( \Phi: (M,g,J) \to (N,\widetilde{g},\widetilde{J}) \) of constant rank between almost Hermitian manifolds w.r.t. the properties named in the title, using the \textit{H.-C. Lee} form \( \alpha \) [Am. J. Math. 65, 433-438 (1943; Zbl 0060.38302)] and its decomposion into horizontal and vertical parts \( \alpha_{h}, \alpha_{v} \). The map \( \Phi \) should be neither constant nor immersive. Under the additional assumption that \( N \) is \( (1,2) \)-symplectic, the main result states that two of the properties: \( \Phi \) is harmonic, \( \Phi \) has minimal fibres, \(\alpha_{h}\) is an \(H\)-annihilator, always imply the third. This generalizes a result of \textit{P. Baird} and \textit{J. Eells} [Geometry, Proc. Symp. Utrecht, Lect. Notes Math. 894, 1-25 (1981; Zbl 0485.58008)] concerning the case that the target is a Riemann surface. Also the subcase of harmonic morphisms is considered. Finally, obstructions to the conformal changes of metric rendering \(\Phi\) harmonic or with minimal fibres are established.