Concerning 2) one can, of course, take the product of two curves of higher genus to get a counter-example. In general, to have a statement as you want, one should look for rigid complex surfaces of general type (with ample $K$), because as soon as such a surface has a deformation, we get a counterexample (by Yau Aubin).

Concerning 3), the situation depends on whether the curvature is zero or not. The universal cover of a manifold of constant holomorphic sectional curvature is either i) $\mathbb CP^n$ with Fubini-Studi metric, or ii) $\mathbb C^n$ with a flat metric or iii) the open unit ball $\mathbb B^n$ with complex hyperbolic metric. See discussion here (don't be mislead by the title :) ):

Kahler manifolds with constant bisectional curvature

In the case i) it is a well known theorem, that a Kaehler manifold diffeomorphic to $\mathbb CP^n$ is in fact byholomoprhic to $\mathbb CP^n$. So the answer to your question is positive.

In the case ii) we can take two elliptic curves with a flat metric of area $0$, that are not byholomorphic. So the answer to the question is negative.

Finally, I believe that the answer is also negative in case iii), provided the dimension is greater than $1$. This is certainly true for complex dimension $2$. Indeed, by Yau's theorem any complex surface diffeomorphic to a ball quotient is in fact byholomorphic to it, and also the complex structure on a ball quotients is rigid.

Concerning 2) one can, of course, take the product of two curves of higher genus to get a counter-example. In general, to have a statement as you want, one should look for rigid complex surfaces of general type (with ample $K$), because as soon as such a surface has a deformation, we get a counterexample (by Yau Aubin).

Concerning 3), the situation depends on whether the curvature is zero or not. The universal cover of a manifold of constant holomorphic sectional curvature is either i) $\mathbb CP^n$ with Fubini-Study metric, or ii) $\mathbb C^n$ with a flat metric or iii) the open unit ball $\mathbb B^n$ with complex hyperbolic metric. See the discussion here (don't be mislead by the title :) ):

Kahler manifolds with constant bisectional curvature

In the case i) it is a well known theorem, that a Kaehler manifold diffeomorphic to $\mathbb CP^n$ is in fact biholomoprhic to $\mathbb CP^n$. So the answer to your question is positive.

In the case ii) we can take two elliptic curves with a flat metric of area $0$, that are not biholomorphic. So the answer to the question is negative.

Finally, I believe that the answer is also negative in case iii), provided the dimension is greater than $1$. This is certainly true for complex dimension $2$. Indeed, by Yau's theorem any complex surface diffeomorphic to a ball quotient is in fact biholomorphic to it, and also the complex structure on a ball quotient is rigid.

Concerning 2) one can, of course, take the product of two curves of higher genus to get a counter-example.

Concerning 3), the situation depends on whether the curvature is zero or not. The universal cover of a manifold of constant holomorphic sectional curvature is either i) $\mathbb CP^n$ with Fubini-Studi metric, or ii) $\mathbb C^n$ with a flat metric or iii) the open unit ball $\mathbb B^n$ with complex hyperbolic metric. See discussion here (don't be mislead by the title :) ):

Kahler manifolds with constant bisectional curvature

In the case i) it is a well known theorem, that a Kaehler manifold diffeomorphic to $\mathbb CP^n$ is in fact byholomoprhic to $\mathbb CP^n$. So the answer to your question is positive.

In the case ii) we can take two elliptic curves with a flat metric of area $0$, that are not byholomorphic. So the answer to the question is negative.

Finally, I believe that the answer is also negative in case iii), provided the dimension is greater than $1$. This is certainly true for complex dimension $2$. Indeed, by Yau's theorem any complex surface diffeomorphic to a ball quotient is in fact byholomorphic to it, and also the complex structure on a ball quotients is rigid.

Concerning 2) one can, of course, take the product of two curves of higher genus to get a counter-example. In general, to have a statement as you want, one should look for rigid complex surfaces of general type (with ample $K$), because as soon as such a surface has a deformation, we get a counterexample (by Yau Aubin).

Concerning 3), the situation depends on whether the curvature is zero or not. The universal cover of a manifold of constant holomorphic sectional curvature is either i) $\mathbb CP^n$ with Fubini-Studi metric, or ii) $\mathbb C^n$ with a flat metric or iii) the open unit ball $\mathbb B^n$ with complex hyperbolic metric. See discussion here (don't be mislead by the title :) ):

Kahler manifolds with constant bisectional curvature

In the case i) it is a well known theorem, that a Kaehler manifold diffeomorphic to $\mathbb CP^n$ is in fact byholomoprhic to $\mathbb CP^n$. So the answer to your question is positive.

In the case ii) we can take two elliptic curves with a flat metric of area $0$, that are not byholomorphic. So the answer to the question is negative.

Finally, I believe that the answer is also negative in case iii), provided the dimension is greater than $1$. This is certainly true for complex dimension $2$. Indeed, by Yau's theorem any complex surface diffeomorphic to a ball quotient is in fact byholomorphic to it, and also the complex structure on a ball quotients is rigid.