The essence of the philosophic mindset involves embracing Socrates' approach of questioning the walls of the boxes one is thinking within.
Pursuit of this approach has lead me to the conclusion that the two dichotomies in your title are self-serving rationalizations by one of the two views in each of your dichotomies.
For irrationalism vs positivism, the actual viewpoint one should be looking at is "the limits to rationalism". One of the limits to rationalism was decisively articulated by Kant in his "The Critique of Pure Reason". The subsequent demonstration of the limits of any closed form logic by Godel further amplified the limits of rationalism.
Rationalists tend to adopt classical logic as the "One True Logic", and classical logic holds by the Law of the Excluded Middle. Logicians have found that classical logic is itself only one of multitudes of possible logics, and most of the others admit to more than two truth states -- IE they allow both true, false and "other" as truth categories, violating LEM. If one holds by classical logic, then the refutation of pure rationalism requires irrationalism. For any advocate of pluralist logics, rationalism can be seen as locally "true" and a useful paradigm, but limited in applicability. The actual alternative to pure rationalism is pluralism, where other epistemologies are also useful for uncovering "truth" to varying degrees in varying circumstances.
So -- Positivism's claim that one must embrace either a positivist approach to epistemology, or irrationalism, is rejected as an invalid claim by most philosophers and educated people. Ther ARE people who assert irrationalism, and yes, those people reject positivism, but those people are themselves taking the other part of the false dichotomy that classical logic invalidly assumes.
Answering your related questions, yes "empiricism and rationalism are two types of knowledge sources in epistemology". Rationalism is an effort to define our world thru reason alone, empiricism is an effort to infer it indirectly thru observations followed by theoretical modeling.
"Are positivism and irrationalism also types of knowledge sources in epistemology"? No. They are global epistemic perspectives. Positivism asserts we can only know with certainties, and irrationalism asserts the inability to support knowledge.
Positivism, rationalism, irrationalism, and empiricism. Positivism falsely claims that empiricism provides certainties, or at least "close enough" certainties. It tries to harness rationalism and empiricism together. Irrationalism is in opposition to both the empirical and rational sources of knowledge and asserts that we have other non-rational and non-empirical knowledge paths.
For pragmatism vs idealism one of your sub questions helps illuminate what is going on here, and where the self-serving claims lie. The subquestion that illuminates this is "Are pragmatism and materialism the same?"
Pragmatism has a positive implication in conventional usage, and materialists have tried to harness that by claiming themselves to be "pragmatic" as opposed to the NON-pragmatic idealists. This is a mis-statement of what pragmatism consists of.
The key principle of pragmatism is the adoption of the pragmatic truth criteria, as opposed to the logical one. The pragmatic truth criteria is that "Truth is what is useful" and this is opposed to the logic criteria that "Truth is an absolute and objective fact". There are clear political and sociological problems with "truth is what is useful", as usefulness is defined by either a community, or by power-brokers within a community. These political and sociological problems have been an obstacle to the widespread embrace of pragmatic truth within philosophy, or admission that it is intrinsic to both philosophy and science. However, Popperian science explicitly embraces pragmatic truth, and Popperian science is how science is done today. This is a bit of a dirty secret --science facts are established by the consensus judgement of expert opinion. Pragmatic truth only works if the custodians of power in the community are trustworthy.
Materialists falsely claim that materialism has been established as true by pragmatic means. This is the self-serving story that applies in your second question. Materialism is actually FALSIFIED by science -- which is a pragmatic enterprise. Both Einstein's relativity, and the Quantum revolution, show that matter is not fundamental. Materialists tend to focus on other aspects of science progress. The success of material reductionism to explain almost all of physics, and about half of chemistry, and the complexity of biochemistry, dramatically undercut a lot of the theories based on idealist and dualist alternatives to materialism. All three ontological perspectives have not fared well against 20th and 21st century science.
This comment will be useful to build on to answer your other sub questions. The two major disciplines within metaphysics are epistemology and ontology. Pragmatism is an epistemological framework, as it has to do not only with how to find truth, but even what truth IS. Rationalism and empiricism and irrationalism are all also epistemic perspectives. Materialism, idealism, and dualism are all ontologies -- they are claims about what is fundamentally real.
Pragmatism and empiricism play well together, and both play very well with indirect realism. Logic and rationalism play well together. Irrationalism -- tests to play best with intuitionism and versions of direct realism. One can arrive at any of the three main ontologies (or multiple variants off the three main ones) from any of these epistemologies.
