memes

Again, for example, I am a mathematical Platonist. What beliefs of mine in that regard contradict with Marxism?

Well for one you see math as something independent, you see math existing in a vacuum.

"Contrary to metaphysics, dialectics does not regard nature as an accidental agglomeration of things, of phenomena, unconnected with, isolated from, and independent of, each other, but as a connected and integral whole, in which things, phenomena are organically connected with, dependent on, and determined by, each other.

The dialectical method therefore holds that no phenomenon in nature can be understood if taken by itself, isolated from surrounding phenomena, inasmuch as any phenomenon in any realm of nature may become meaningless to us if it is not considered in connection with the surrounding conditions, but divorced from them; and that, vice versa, any phenomenon can be understood and explained if considered in its inseparable connection with surrounding phenomena, as one conditioned by surrounding phenomena." (Dialectical and Historical Materialism)

This is my understanding of your views from what you wrote in other replies, i haven't read and don't plan to read about mathematical Platonism, we already have developed much further than whatever Plato worked with.

The production of ideas, of conceptions, of consciousness, is at first directly interwoven with the material activity and the material intercourse of men, the language of real life. Conceiving, thinking, the mental intercourse of men, appear at this stage as the direct efflux of their material behaviour. The same applies to mental production as expressed in the language of politics, laws, morality, religion, metaphysics, etc., of a people. Men are the producers of their conceptions, ideas, etc. – real, active men, as they are conditioned by a definite development of their productive forces and of the intercourse corresponding to these, up to its furthest forms. Consciousness can never be anything else than conscious existence, and the existence of men is their actual life-process. If in all ideology men and their circumstances appear upside-down as in a camera obscura, this phenomenon arises just as much from their historical life-process as the inversion of objects on the retina does from their physical life-process.

In direct contrast to German philosophy which descends from heaven to earth, here we ascend from earth to heaven. That is to say, we do not set out from what men say, imagine, conceive, nor from men as narrated, thought of, imagined, conceived, in order to arrive at men in the flesh. We set out from real, active men, and on the basis of their real life-process we demonstrate the development of the ideological reflexes and echoes of this life-process. The phantoms formed in the human brain are also, necessarily, sublimates of their material life-process, which is empirically verifiable and bound to material premises. Morality, religion, metaphysics, all the rest of ideology and their corresponding forms of consciousness, thus no longer retain the semblance of independence. They have no history, no development; but men, developing their material production and their material intercourse, alter, along with this their real existence, their thinking and the products of their thinking. Life is not determined by consciousness, but consciousness by life. In the first method of approach the starting-point is consciousness taken as the living individual; in the second method, which conforms to real life, it is the real living individuals themselves, and consciousness is considered solely as their consciousness.

This method of approach is not devoid of premises. It starts out from the real premises and does not abandon them for a moment. Its premises are men, not in any fantastic isolation and rigidity, but in their actual, empirically perceptible process of development under definite conditions. As soon as this active life-process is described, history ceases to be a collection of dead facts as it is with the empiricists (themselves still abstract), or an imagined activity of imagined subjects, as with the idealists.

Karl Marx - The German Ideology
https://www.marxists.org/archive/marx/works/1845/german-ideology/ch01a.htm

My understanding of this passage is that in idealist thinking, ideas shape reality but in materialist thinking, material reality shapes ideas.

Look into historical materialism. Marx's materialist method of understanding how human societies evolved hinged on giving primacy to factors like energy, production, population and ecology. It allowed him to construct an understanding of anthropology so advanced for his time that when I was being taught anthropology by my uni professors I legit thought he was a Marxist. Only later did I realise that marx's method is today being rediscovered and being touted as some new revolutionary thinking.

Another point in which materialism is important to Marxism is with economics. Whole today's economic theories on value consider it to be subjective, marx analysed value through constraints on labour in an economy. The method that he used was a kind of primitive linear programming. It then inspired the creation of actual linear programming, which won Nobel prizes and forms the backbone of economic planning even in capitalist firms.

Really, marx's dialectical materialism was one of the first scientific approaches to fields and political movements (economics, history, socialism) dominated by idealism and hodgepodge theories. It is the reason why marx was so influential beyond his years and beyond his contemporaries like proudhon and fruerbach

Does Marxism draw any [relevant] conclusions from materialism?

Of course. One of the tenets of Marxism is that social being determines social consciousness, not vice versa. This is textbook materialism. Anything else simply leads to liberalism, religion, etc.

the idealist schools of thought that only recognise material and non-material mental stuff

You seem confused. There are two main currents in philosophy - materialism, which posits that nature is primary to spirit, being is primary to thought, and idealism, which posits the opposite. In between the two there are agnostics, who claim this question is impossible to answer. This has nothing to do with whether a philosophy "recognizes non-material stuff".