Whenever somebody says something, one has to consider their Competency, Context and Time to read between the lines and try and get at the gist of what is being intended rather then the mere words themselves.

This is orders of magnitude more important when we try to understand "The Greats" (in any field) because they have a certain insight into things, a huge knowledge base to draw upon and are aware of myriads of nuances involved in their domain of expertise all of which we poor students have to struggle to grasp.

It is with the above points in mind that one should interpret Knuth's nuanced quote above. He had the highest of regards/respect for Dijkstra (see his detailed review (pdf) of Dijkstra's book "Structured Programming" at http://infolab.stanford.edu/pub/cstr/reports/cs/tr/73/371/CS...) Given that Donald Knuth is the "Patron Saint of Yak Shaves" (https://yakshav.es/the-patron-saint-of-yakshaves/) his perspective did not allow a fully mathematical abstraction without any reference to a machine (real or virtual). Note that he did not deny Dijkstra's approach but merely wasn't agreeing with Dijkstra's strident promotion of it. That is a viewpoint and not a value judgement.

Coming back to the points above;

[1-3] - One has to remember that when Dijkstra did these almost nothing that we take for granted today existed. Everything had to be built from ground-up. You can imagine the difficulty when you have nothing/nobody to learn from and are basically inventing stuff as you go along. You only had Mathematics as a stable bedrock.

[4] No, he pioneered it. Wikipedia actually credits him with coining the very term "Structured Programming" (https://en.wikipedia.org/wiki/Structured_programming#cite_no...) Regardless, he wrote EWD249 "Notes on Structured Programming" in 1969. Also see EWD1308 "What led to Notes on Structured Programming".

[5] Most people misunderstand what is meant by "Correctness" in a Computer Program. Perhaps Bertrand Meyer's synonym of a "Contract" is better suited to explain that what is meant is "Correctness w.r.t. Specification". Obviously this is fundamental to Programming itself. To get the benefit of this approach you don't have to go 100% rigour (rather difficult) but train yourself to think in these terms and using appropriate pre/post conditions. A good example can be seen in the use of Design-by-Contract syntax in Eiffel. Here is Dijkstra from EWD288 "Concern for Correctness as a Guiding Principle for Program Composition"

My thesis is, that a helpful programming methodology should be closely tied to correctness concerns. I am perfectly willing to admit that I myself may be the most complete victim of my own propaganda, but that will not prevent me from preaching my gospel, which is as follows. When correctness concerns come as an afterthought and correctness proofs have to be given once the program is already completed, the programmer can indeed expect severe troubles. If, however, he adheres to the discipline to produce the correctness proofs as he programs along, he will produce program and proof with less effort than programming alone would have taken.

Finally, a word or two about a wide-spread superstition, viz. that correctness proofs can only be given if you know exactly what your program has to do, that in real life it is often not completely known what the program has to do and that, therefore, in real life correctness proofs are impractical. The fallacy in this argument is to be found in the confusion between "exact" and "complete": although the program requirements may still be "incomplete", a certain number of broad characteristics will be "exactly" known. The abstract program can see to it that these broad specifications are exactly met, while more detailed aspects of the problem specification are catered for in the lower levels.

[6-7] Your comment "Notably, none of his accomplishments in 1-3 were attained according to the methodology he advocated later in life as the only possible one" is exactly the wrong inference to draw! It is precisely due to this formative experience that he was such a strident proponent of the use of Mathematics in Programming. As pointed out above, he had to invent a lot of this stuff as he went along and had nothing other than mathematical logic to aid him in his endeavour. Dijkstra actually said he was "slow" to understand the works of Floyd/Hoare/Others on Program Correctness techniques. But of course once he got involved, he took it to the next level.

In summary, read Dijkstra, try and get at what he intended to convey rather than the tone/words (which are irrelevant in the broader scheme of things) and put forth effort to understand what has been said/written down. The rewards are great.