dg.differential geometry - De Rham decomposition theorem, generalisations and good references

I suppose we have to ask the Riemannian manifold to be complete. Otherwise $mathbb R^3 - lbrace 0 rbrace$ would be a counterexample.

I do not have an answer to question 2, but you might be interested in variations of De Rham decomposition Theorem to the realm of compact complex manifolds. There, it is natural to ask, as Beauville did, if a holomorphic decomposition of the holomorphic tangent bundle implies that the universal covering is isomorphic to a product(no metric assumption).

Without further assumptions there is no hope since Hopf surfaces provide examples with decomposable tangent bundle but with universal covering isomorphic to $mathbb C^2 - lbrace 0 rbrace$.

If one assume that the manifold is projective or Kahler then there are some positive results, the first of which can be found in Beaviulle's paper linked to above. In the projective case you can also look at this, this, and this paper. In the Kahler case you can look at here.

The general problem seems to be wide open, in the above results either one
assumes that one of the factors is one-dimensional or imposes strong conditions on the ambient variety itself (dimension $le 3$ or uniruledness).