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 and the Wikipedia article 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.

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 and the Wikipedia article 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.

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.

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 and the Wikipedia article 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.