Let $X$ be a contractible 2dimensional simplicial complex. Are there nice necessary and sufficient conditions for $X$ to be embeddable in $\mathbb R^3$? Clearly it is necessary that the link of every vertex be a planar graph. Is this sufficient?

If your complex is finite, then figure out the possible ways of thickening it to a 3manifold. The possible thickenings are determined by the various embeddings of the links of the vertices into $S^2$, then seeing if these induce compatible thickenings over the edges (determined by the same cyclic ordering over the link of the edge) and faces of the complex. If it can be thickened this way, then it must be a ball since it is a contractible 3manifold. 

