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 and the Wikipedia article on the Whitehead manifold for some closely related material.