What is known about spaces of embeddings of contractible manifolds into Euclidean space? I am also curious about the case of small codimension (or even codimension 0). The same question about the configuration spaces in such manifolds.

(This is by far not a complete answer, just an example.) In dimension 4, a paper of Livingstone (build on previous work of Lickorish) constructs some (compact with boundary) contractible 4manifold which embeds in $\mathbb R^4$ in infinitely many (countable) distinct ways. These are distinguished by the fundamental group of the complement. 

