Please allow a correction: I think your statement "he says that one should sincerely try to disproof hypotheses – and I am quite certain that he didn’t mean the null hypothesis that Fisher formulated but rather the hypothesis that is of critical importance to us" is not really correct. Actually, this is exactly how Fisher would have described "null hypothesis". For Fisher, there is only one type of hypothesis: the hypothesis to be tested, a.k.a. the null hypothesis. The distinction between null and alternative hypothesis, introduced by Neyman and Pearson, has never been accepted by Fisher (also Conifold's statement "Instead of testing the actual hypothesis directly Fisher suggested testing its 'negation', the null hypothesis, if the null hypothesis comes out as unlikely after the test then our actual hypothesis 'survived the test'" does not describe Fisher's approach correctly)
If you read Fisher carefully, the commonly used version of "null hypothesis" (nil, no effect, "just random variation" and similar) is not what Fisher was referring to. The central meaning of a test for statistical significance is its ability to create results which can be, following Fisher ("The Design of Experiments", 2nd ed., 1937), divided into two classes with different interpretations:
“those which show a significant discrepancy from a certain hypothesis; […] and […] results which show no significant discrepancy from this hypothesis. This hypothesis […] is again characteristic of all experimentation. […] [W]e may speak of this hypothesis as the “null hypothesis”, and it should be noted that [it] is never proved or established, but is possibly disproved, in the course of experimentation. […] If it were asserted that the subject would never be wrong in her judgements we should again have an exact hypothesis, and it is easy to see that this hypothesis could be disproved by a single failure, but could never be proved by any finite amount of experimentation. It is evident that the null hypothesis must be exact, that is free from vagueness and ambiguity, because it must supply the basis of the ‘problem of distribution,’ of which the test of significance is the solution. A null hypothesis may, indeed, contain arbitrary elements, and in more complicated cases often does so.” (p.18-20; emphasis added)
Please note that the only distinctive feature of a ‘null hypothesis’, as it is characterized by Fisher, is its ability to partition the results of an experiment into two mutually exclusive classes (supporting and contradicting cases), and to do so, it needs to be exact. That's all. It is not assumed that a null hypothesis has to state the absence of any effect, as most statistical resources – websites, articles and textbooks – claim. It is also important that, as already mentioned, in Fisher's approach there is no "alternative" hypothesis, which is frequently equated with the "research hypothesis" to be supported by rejecting the null. In Fisher's approach, the "null hypothesis" is the hypothesis the researcher wants to test. It can state the absence of an effect or its existence, and as long as it is exact, it can be tested. However, to test a non-nil null hypothesis appropriately, the usually applied tests (e.g. t-Tests) have to be replaced by versions that reflect the effect (size). In the case of the t-Test, any effect could be tested using a particular noncentral version of this test (which uses the same distribution that is used for power calculations in the Neyman-Pearson approach). A significant p would - as usual - indicate that the data obviously do not correspond with the prediction based on the null hypothesis (our research hypothesis!), which would commonly be considered as rejection of the null. Interpreted this way, at least the majority (if not all) of the usually discussed flaws of "NHST" disappear. Moreover, this interpretation of significance testing looks like a statistical version of Popper's falsification principle, or at least as a statistical argument closely related to it.
The confusion of many users of statistical methods (which has also driven the - still ongoing - discussion about NHST, "null hypothesis significance testing" in the "soft" or - better - weak "sciences" [like Psychology]) is probably due to mixing up two distinct approaches to testing hypotheses - Fisher's significance test on one side and Neyman-Pearson's theory of statistical decision on the other - into an "inconsistent hybrid that every decent statistician would reject" (Gigerenzer, 1993). A prototypical study, at least in Psychology, works like this: The researcher has an assumption, call it A, that there is some effect. He/she assumes a medium effect size (Cohen's d=0.5; poor theory, probably, but anyway...) and calculates the sample size for this assumed effect to be indicated sensitively, say with Power=0.8 (this is a kind of Neyman-Pearson). Then he/she collects data, runs a standard (central) t- or F-Test, setting a strawman null hypothesis of "no effect" and if p<0.05, he/she rejects the null (which is ok; this is Fisher, but not testing the actual hypothesis) and accepts A (which is not ok, as any effect size different from null gets support from rejecting the null, unless all other alternatives can be ruled out). This latter conclusion is neither Fisher, nor Neyman-Pearson, it is simply not correct.
To sum up, I think that the abstract you are referring to is rather one of the rare cases in which the essence of testing hypotheses (in a way that would have been accepted by Fisher) has been extracted more or less correctly. Still, there is some inexactness in the abstract, since it assumes that the null and the research hypothesis are different things. But actually, for both, Popper and Fisher, it is the research hypothesis that needs to be tested and, if necessary, rejected. Which actually is Fisher's null hypothesis.
With regard to your third question:
In "Logik der Forschung" Popper sometimes refers to Fisher's likelihood concept, but not to significance testing (or to the Neyman-Pearson theory). In Bennett's (1990) "Statistical Inference and Analysis - Selected Correspondence of R.A. Fisher", Popper is not listed as correspondent.