Are there Riemanniann manifolds with zero curvature other than open subsets of $\mathbb{R}^n \times \mathbb{T}^m$, where $\mathbb{T}^m$ is an $m$ dimensional torus and $m,n\geq 0$ ?

Does taking quotients of opens in $\mathbb{R}^n \times \mathbb{T}^m$ by the action of some -possibly discrete- Lie group provide new examples (i.e. that are not themselves isometric to opens in a Torus x Euclideanspace)? [assuming we're in a case where the quotient is a manifold, and we can induce a metric on it]

Does taking (universal or not) coverings enlarge the class of the above examples?

More generally: is there a classification of flat not necessarily complete flat Riemannian manifolds?

If we added "complete", would we obtain only the $\mathbb{R}^n \times \mathbb{T}^m$ 's ?