No. Take the torus $T^2=\mathbb R^2/\mathbb Z^2$ and consider the self-map induced by the matrix

$$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$

No. Take the torus $T^2=\mathbb R^2/\mathbb Z^2$ and consider the self-map induced by the matrix

$$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$

PS. As for simply-connected examples, I think that a quintic in $\mathbb CP^3$ is an example. One should take any self-diffeo $\varphi $ of such a quintic that has infinite order in the mapping class group. I guess such a diffeo can be constructed as a product of Dehn twists - induced from the family of all non-singular quintics in $\mathbb CP^3$. Now, such $\varphi$ can not be induced by a $J$-holomorphic map, because any complex surface diffeomorphic to a quintic is of general type (because Kodaira dimension is a diffeo invariant in dimension 4). Finally, any variety of general type has a finite group of holomorphic automorphisms.

No. Take a torus $T^2=\mathbb R^2/\mathbb Z^2$ and consider the self-map induced by the matrix

$$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$

No. Take the torus $T^2=\mathbb R^2/\mathbb Z^2$ and consider the self-map induced by the matrix

$$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$