The answer to 2 is yes, there is such an example. In
>McMillan, D. R., Jr., *Some contractible open $3$-manifolds*. Trans. Amer. Math. Soc. **102** (962), 373--382.

there is a construction of uncountably many topologically distinct, contractible (open) $3$-manifolds $M_\alpha$ such that $M_\alpha \times \mathbb R$ is homeomorphic to $\mathbb R^4$.

Take a look at [this recent MO question][1] and the [Wikipedia article][2] on the Whitehead manifold for some closely related material.

[1]: http://mathoverflow.net/questions/60113/contractible-manifolds
[2]: http://en.wikipedia.org/wiki/Whitehead_manifold