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** (1962), 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.

**Edit.** The answer to 3 is also yes, assuming by "torus" you mean $S^1$. On page 221 of
>Vogt, E., *Foliations of codimension $2$ with all leaves compact on closed $3$-, $4$-, and $5$-manifolds*. Math. Z. **157** (1977), no. 3, 201–223.

you can find a construction of infinitely many pairwise non-homeomorphic closed Seifert 3-manifolds whose product with $S^1$ gives the same Seifert 4-manifold.

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