Popper himself believed that Darwin does not fit his criterion, but that it appears to be useful scientifically. Today it is basically invaluable to several important subfields of biology. He admits this cannot be explained on his proposed basis, and is not crushed. (So your answers are 'yes', 'no', and 'yes'. To the extent it is falsifiable, it has been disproven by his own observation. And to the extent he kept supporting it as valuable anyway, it must not have been meant to be falsifiable.)
So, evidently, the proposition has exceptions, and there are other ways of providing scientific value, even in Popper's own estimation.
Lakatos, coming immediately after Popper, emphasized that Popper's criterion is good in spirit, but that theories can abide a certain limited range of exceptions as long as nothing truly better and equally simple comes along. In fact, he gave historical examples of theories that continue to protect themselves from known counterexamples almost indefinitely, without becoming truly compromised. There is a difference between admitting a range of exceptions and admitting tortured constructions that continually rescue the theory.
Popper's main objection was to things like Marxism and Freudianism or religious impositions of Idealism on physics. We can accept his criterion as a 'smell' that makes us look for these distortions without simplifying the procedure of science into a mechanism that uses clean, potentially falsifiable hypotheses as its lynchpin.
To my mind, we need to see Popper's criterion itself in slightly less rigid terms. It has become something one teaches dogmatically in schools, which is sad. Because it has come to pass that the view is held more tightly by Popper's adherents, including too many teachers of basic science, than Popper seems to have held it himself.
Elaboration of personal theory
My approach is to relativize Kuhn's notion of productive periods, and to apply Popper as a criterion for the 'period' he refers to as normal science, and something more like Lakatos to the 'period' he refers to as revolution (leaving someone like Feyerabend to rule over pre-science...).
But each discipline is actually made up of sub-disciplines, and each of those has definite periods of its own, where its relationship to the larger theories of its embracing discipline can be in a posture which is 'prescientific' (Whatever we are doing works, so it must fit with their work, but we don't really care how the two harmonize, because we are happy for now), 'revolutionary' (Their theories challenge our groundwork, and we create a succession of bridges to or walls around our work) or 'normal' (We leave off developing our own internal theories and solve specific gaps between our sub-domain and the larger theory).
This nesting of smaller and smaller subdomains like a kind of self-similar fractal scaling, extends down to the day-to-day operations of real scientists. They might successively, on an hour-by-hour basis ignore discrepancies, bridge or isolate from them, or work out details in hopes of resolving them.
If you reach a point at any level of reconciliation with your including domain where transition back into a normal/'Popper' mode is impossible, and you are stuck in a 'revolutionary' mode, you are either still in a real (mini-) revolution, or you are not really in a 'revolutionary' mode at all, but in a pre-scientific one -- you are keeping your faith and no longer doing science.
If you are permanently stuck in a normal/'Popper' mode, your science is dead and you are doing engineering.
As change becomes quicker, we find ourselves in 'revolutionary' periods of great length or recurring with great frequency. To the degree Popperianism wants to put the brakes on this, it will thwart progress.
I would contend that the 'smelly' non-Popperish theories that rescue themselves instead of accounting risk have this quality, they begin as revolutions and retreat to applying puzzle-solving skills to a set of idees fixes which lets them backslide into permanently pre-scientific devotion.
To me this is the right way to view Popper: as the criterion for when science is being 'normal' in Kuhn's sense along with a dedication to keeping it Kuhnian 'normal' a norm, though not a rule.