After Lukas editing his question, and by his comments too, I realize it will be difficult to make him see what is accessory and what is the core in the criticism of Popper. I leave to others to do so. It is repeated too by many that this site is not a forum for debate. Therefore now I edit my answer and I will stick only to the main core question:
"Is Poppers Solution to the Problem of Induction still valid?"
The issue here will be the main criticism of the Popper's solution to the problem of induction.
Popper said that induction is not justifiable. That a theory has been corroborated in the past "says nothing whatever about future performance." Popper said that it is possible to avoid assuming that the future will, or probably will, be like the past, and this is why he has claimed to have solved the problem of induction. We do not have to make the assumption, he tells us, if we proceed by formulating conjectures and attempting to falsify them. He says that, as a basis for action, we should prefer "the best-tested theory." Popper never adequately defined the notion of severity for tests, a concept on which much depended, since the more severe the test a theory passed, the better its corroboration. This can only mean the theory that has survived refutation in the past; but why, since Popper says that past corroboration has nothing to do with future performance, is it rational to prefer this? Corroboration is not another term for confirmation since it does not involve any notion of inductive support for a theory. Theories remain as unsupported hypotheses or conjectures forever.
Corroboration is not a measure of verisimilitude. Saying that the better corroborated theory is also the one that is closer to truth would be no more than a guess.
Why is it rational to act on the basis of a decision informed by the best tested and corroborated theory, to apply it to new situations, to decide to use it as basis for practical action? Corroboration says absolutely nothing about the future performance of a theory. In what sense, therefore, is the decision to act a rational one? The reply of Popper is that since it is the best theory, what could be more rational than acting on such a theory, than holding a “pragmatic belief in the results of science”. This reply is not entirely satisfactory. For under the circumstances, the rational thing to do is not to act at all. If our best theory provides us with no clue as to the prospect of achieving our goals, then it cannot sufficiently motivate us to act. For our best theory to guide us in our actions, its past success should give us some reason for its future success. In short, Popper must allow for inductivism. The concept of corroboration cannot explain why it is rational for scientists to base their future predictions on the best corroborated theory. To do this, it is inevitable for them to accept some kind of principle of induction. Without the inductive assumption, the fact that a theory was refuted yesterday is quite irrelevant to its truth-status today. Corroboration is also uncertain and can never be quantified by degree of probability.
Wesley Salmon in his paper “Rational Prediction” focuses attention on the practical case in which one must decide on a course of action on the basis of a theory. Salmon asks how one is to choose between alternative theories which make conflicting predictions as a basis on which to act. According to Popper, the action should be based on the most highly corroborated of the competing theories. But this suggests that corroboration has inductive force. For while corroboration relates to a theory’s past success in surviving tests, if it is to serve as a basis for future action then past survival of tests must be of relevance to what will take place in the future. But if corroboration is to be taken into account in determining a future course of action, this amounts to an inductive inference from past success in surviving tests to the likely continuation of such success into the future. Again, it therefore appears that Popper’s falsificationist philosophy of science rests at base on an assumption that is inductive in nature.
Popper’s theory of method suggests that theories are to be rejected the moment they entail a false prediction. Lakatos denies that there are critical tests, in the Popperian sense, in science. Ruthless elimination of theories does not appear to be the norm in actual science. The point here is that the ‘falsification/corroboration’ disjunction offered by Popper is far too logically neat: non-corroboration is not necessarily falsification, and falsification of a high-level scientific theory is never brought about by an isolated observation or set of observations. Such theories are, it is now generally accepted, highly resistant to falsification. They are falsified, if at all, Lakatos argues, not by Popperian critical tests, but by research gradually grinding them to a halt. Popper's distinction falsifiability does not in the end do full justice to the fact that all high-level theories grow and live despite the existence of anomalies which are incompatible with the theories. The existence of such anomalies is not usually taken by the working scientist as an indication that the theory in question is false; on the contrary, he will usually, and necessarily, assume that the auxiliary hypotheses which are associated with the theory can be modified to incorporate, and explain, existing anomalies. Philosophers of science who hold that the actual practice of science is of relevance to the normative methodology of science will be little inclined to adhere to the Popperian picture in the face of historical evidence of anti-falsificationist practice in science.
To Martin Gardner, every falsification of a conjecture is simultaneously a confirmation of an opposite conjecture, and every conforming instance of a conjecture is a falsification of an opposite conjecture. If Popper bet on a certain horse to win a race, and the horse won, you would not expect him to shout, "Great! My horse failed to lose!". For Popper, the more tests for falsification a theory passes, the more it gains in "corroboration”. It's not so much that Popper disagreed with inductivists as that he restated their views in a bizarre and cumbersome terminology.