Are pairs of events characterized by spacetime intervals even if the spacetime containing these events is not flat? In Wikipedia, spacetime intervals are presented explicitly under the heading "Spacetime intervals in flat space"; apparently including a presentation of spacetime intervals for (all) pairs of events in flat spacetime. 
But is it correct and understood that spacetime interval values, $s^2$, can also be attributed (unambiguously, up to a common non-zero factor) to (all) pairs of events in a (any) spacetime which is not flat?
 A: 
is it correct and understood that spacetime interval values, $s^2$, can also be attributed (unambiguously, up to a common non-zero factor) to (all) pairs of events in a (any) spacetime which is not flat?

No. There can be multiple paths between the same events. And while each path can be broken into small sections that have a well defined interval and you could take the square root of each section of them and add them all up (getting a proper length of the path) and then take the square you will get different answers for the different paths.
And if you try to limit your paths to geodesics, some events have no geodesics between them and others have multiple geodesics between them with different proper lengths for different geodesics. (The proper length is the sum of the square roots of the $s^2$ for each little piece of path.)

for all all-timelike paths I'd evaluate the plain extremum

Consider the set $$\{(v,w,x,y,z)\in\mathbb R^5: v^2+w^2=1\}$$ and give the larger space the metric $ds^2=dz^2+dy^2+dx^2-dw^2-dv^2.$ This induces an obvious metric on the 4d spacetime as an isometrically embedded surface. Then there are arbitrarily long timelike curves and arbitrarily short timelike curves between any two events, I don't even know what you could mean by a plain extremum.
