It would take a book, perhaps several books, to answer all your questions! What any or all of these systems of thought consist in are matters of scholarly dispute and contention. The chapter on Marx in my Freedom’s Progress? is an attempt to give what I think are the basic elements of his thought. Here are some brief excerpts from that chapter:
*All things considered, one very important fact to keep in mind when considering Marx’s influence on others is the obvious but sometimes neglected fact that Marx was a communist!
*Communism would end all human suffering is the claim that ‘the future communist world would be a post-scarcity world. All economic problems would fade away and there would be no need to address the question of the allocation of scarce means among competing needs.’
*Much of the power of Marxism comes from its being an ersatz religion.
*In this twenty-first century, which succeeds a twentieth century that was dominated in many ways by Marxism in various forms, some strong, some weak, it is salutary to remember that, in his lifetime, Marx was virtually unknown and his writings likewise.
*The two most prominent pieces in the Marxist canon are The Communist Manifesto and Capital. Capital has been described as a ‘massive, rancorous and obscure book’ but ‘less unreadable than its structure and opening chapters would suggest.’ What, if anything, is new in Capital? What will we find there that isn’t already contained in the earlier writings of Marx? Very little, unless it’s the development along economic lines of a pre-reflective moral indignation at the widespread poverty of the nineteenth century, inserted into a theoretical economic frame which, based on a misguided concept of value, simply can’t do the work it’s supposed to do.
*The motive force of Marx’s writing was a deep revulsion at certain aspects of modern life. His passion did not arise from any concern with abstract economics but from a visceral rejection of the exploitation he believed to be the inevitable accompaniment to capitalism. His study of economics was meant to explain this exploitation theoretically and practically to undermine it. It is vitally important to understand that Marx’s objection to capitalism was not that it did not work; on the contrary, it worked all too well! If exploitation was the core topic of Marx’s mature and late work, alienation was the topic of his earliest writings. The works of Marx, then, can be divided into two groups: the early ones dealing with the notion of alienation and the later ones concerning themselves with exploitation. The notion of alienation comes to Marx from Hegel and is relayed by Marx to the twentieth century in the form of the disdain of the Frankfurt School thinkers for the ordinary man and his desires. The idea of exploitation lies at the heart of Marx’s mature work and the explanation of this phenomenon is what Marx’s economic thinking is all about.
I recommend Kolakowski’s Main Currents of Marxism (several volumes) which is a classic in the field. You should be able to find this in a good library or, if you’re lucky (as I was) very inexpensively in a second-hand bookshop.
Communism vs Socialism? Well, to be brief, Socialism is Communism-lite! Instead of just appropriating your property outright, Socialists impose a variety of taxes, as much as they can get away with, which will have much the same practical effect even if the legal regime still permits the ownership of property. So, a super-tax rate of 95%! (The spur for the Beatle’s song, Taxman!), or inheritance taxes, or VAT (which is a tax on the money you have left after they’ve extracted income tax), etc. Both Communism and Socialism are inspired by envy and resentment and differ in the end only inasmuch as one is relatively honest about what it’s doing and the other disguises its purpose for strategic reasons.
Finally, Marx and Hegel. Once again, a very complicated and much-disputed question but, in brief, Marx began his intellectual life as a Hegelian, taking from Hegel the basic idea that dynamic change, not stability, was the nature of social and political reality. He took over Hegel’s dialectical philosophy of history and interpreted it in economic terms.
I hope this helps somewhat but there is really no short answer to your excellent questions!
All the best,