Yes. This is the E. Hopf maximum principle from 1927

Right, so the answer is the maximum principle applied to an operator with the same principal symbol as the Laplacian (The difference is a first-order term coming from the brackets [X,JX] which are non-zero in general). About the local question, I agree that if there are n independent functions around a point (dim M=2n), then the J is integrable around that point. Probably, if the J satifies further restrictions (e.g, nearly Kähler in dimension 6), then the existence of ONE non-constant holomorphic function would already imply the integrability. But this is another question.

A holomorphic function is one that satisfies the Cauchy-Riemann equations or, equivalently, whose derivative vanishes on the (0,1)-component of the complexified vector space.