The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. But for example, if a complete Riemannian manifold $M$ satisfies one of the following, can I conclude that $M$ is a $K(G,1)$?

$M$ has sectional curvature bounded above by some negative number.

$M$ has negative sectional curvature.

$M$ has nonpositive sectional curvature.

$M$ has sectional curvature bounded above by $f(\operatorname{vol}(M))$ (where $f: \mathbb R \to \mathbb R$ is some function depending only on the dimension of $M$ that I don't know).

$M$ has scalar curvature bounded above by some negative number.

$M$ has negative scalar curvature.

$M$ has nonpositive scalar curvature.

$M$ has scalar curvature bounded above by $f(\operatorname{vol}(M))$.

Do the answers change if I assume that $M$ is compact? Have I left out a relevant condition of some kind?