At least in the projective setting the following holds true (this is taken from J. Kollár "Shafarevich maps and automorphic forms", Proposition 13.14.2).

Proposition. Let $X$ be a smooth projective variety. If $K_X$ is nef but not big, and $X$ has generically large fundamental group, then $\chi(\mathcal O_X)=0$.

Here, generically large fundamental group roughly means that $\operatorname{Im}\bigl(\pi_1(W)\to\pi_1(X)\bigr)$ is infinite for every positive dimensional subvariety passing through a very general point of $X$. In particular, your hypothesis 2) implies that $X$ has generically large fundamental group, in a strong way.

Thus, if $\chi(\mathcal O_X)\ne 0$, $X$ is projective, and your hypothesis 2) holds true, it follows that $K_X$ must be big and nef.

Observe that (as I wrote in one of the comments) it is nef not because it is big! It is nef because there cannot be rational curves in $X$, as abc remarked in another comment.

I really don't see how Kollár's proof of the proposition above might be transposed to a non projective setting.

At least in the projective setting the following holds true (this is taken from J. Kollár "Shafarevich maps and automorphic forms", Proposition 13.14.2).

Proposition. Let $X$ be a smooth projective variety. If $K_X$ is nef but not big, and $X$ has generically large fundamental group, then $\chi(\mathcal O_X)=0$.

Here, generically large fundamental group roughly means that $\operatorname{Im}\bigl(\pi_1(W)\to\pi_1(X)\bigr)$ is infinite for every positive dimensional subvariety passing through a very general point of $X$. In particular, your hypothesis 2) implies that $X$ has generically large fundamental group, in a strong way.

Thus, if $\chi(\mathcal O_X)\ne 0$, $X$ is projective, and your hypothesis 2) holds true, it follows that $K_X$ must be big and nef.

Observe that (as I wrote in one of the comments) it is nef not because it is big! It is nef because there cannot be rational curves in $X$, as abc remarked in another comment.

I really don't see how Kollár's proof of the proposition above might be transposed to a non projective setting.

Addendum (after the comments of Jason Starr and YangMills). There is a result by J. McKernan which confirms a conjecture by J. Starr (you can find a link to that paper in the comment of YangMills below) roughly stating that the conclusions of the bend-and-break still hold in the setting of $\mathbb Q$-factorial proper algebraic spaces. Over $\mathbb C$, a smooth algebraic space is the same as a Moishezon manifold, i.e. a complex manifold whose space of meromorphic functions has transcendence degree equal to its dimension.

So, if you know in advance that for some reason your $X$ is a little more than merely a compact complex manifold, namely if it is a compact Moishezon manifold (in this compact case, this is equivalent to be bimeromorphic to a projective manifold), then if $K_X$ is not nef you get some rational curve and hence some compact submanifold of the universal cover of $X$. This is not possible by your assumption 2).

Summing up, your assumption 2) implies that $K_X$ must be nef, unless possibly if $X$ is not bimeromorphic to a projective manifold. Of course if $K_X$ is big then $X$ is bimeromorphic to a projective manifold. But still, even with your assumptions 1)+2), I am not able to prove the bigness of $K_X$ if $X$ is only assume to be compact complex, or even compact Moishezon.