My question is about embeddability of 3-dimensional complexes in R^3. Do we have something like Kuratowski's theorem for complexes in 3-space which specifies a set of minors for non-embeddability?
No, in higher dimensions there is no analogue of Kuratowski's theorem. See thesis of Anna Gundert leisurely overview, and the references. There is also a recent work of Matoušek and Wagner that addresses how to determine if a complex is embeddable. They have a nice table for different dimensions on page 4 from it transpires it is not known how hard the problem is.