As asked Dusan Repovs (who is an expert in the theory of topological manifolds), and he sent me the following answer:

This is indeed best possible result, since whenever a product of two spaces is a topological manifold, both factors must be generalized manifolds - which in dimensions below 3 are topological manifolds.

Ref.: A.Cavicchioli, F.Hegenbarth and D.Repovš, Higher-Dimensional Generalized Manifolds: Surgery and Constructions, EMS Series of Lectures in Mathematics, European Mathematical Society, Zurich, 2016.

To the second question: there are also compact examples (in dimensions >3): e.g. take the product of the 3-sphere $S^3$ modulo the Fox-Artin wild arc $A$ and $S^1$. This product is homeomorphic to $S^3\times S^1$.

Ref.: R.J.Daverman, Decompositions of Manifolds, Academic Press, Orlando, 1986.

**Added in Edit:** Answering a comment of John Samples, Dusan Repovs pointed out that the Chapter 29 of Daverman's book contains the following fact: for any $n,m>2$ there are a generalized $n$-manifold $X$ and a generalized $m$-manifold $Y$ which are not topological manifolds, but their product $X\times Y$ is a topological manifold.