Is there something like the Chow lemma for complex spaces (stating that every compact complex space is birational to a projective variety, or some variant of this, like: every proper morphism of complex spaces is birational to a projective morphism)?

As pointed out in the comments, if a complex compact manifold $X$ of dimension $n$ is bimeromorphic to a projective variety, its field of meromorphic functions must have transcendence degree $n$ (the maximum possible)  one says that $X$ is a Moisezon manifold. Conversely, a deep theorem of Moisezon asserts that any compact Moisezon manifold becomes projective after a finite number of blowing ups with smooth centers. 

