I have regularly heard debate between anarcho-socialists and anarcho-capitalists about which one is the true oxymoron.
Well according to the history of that ideology:
The anti-capitalist versions is the original one and preceded the Ancap term by about 100 years. Even when considering the difference between individualist anarchists and collectivist anarchists, neither would be too fond of the anarcho-capitalist and vice versa:
While anarcho-capitalism is sometimes described as a form of individualist anarchism, many others disagree with this as individualist anarchism is largely socialistic. Murray Rothbard, the founder of anarcho-capitalism, argued that individualist anarchism is different from anarcho-capitalism and other capitalist theories due to the individualist anarchists retaining the labor theory of value and socialist economics.
They operate with radically different definitions of what anarchism is, "classical anarchism" rejects any social hierarchy where people subjugate others and aim to dismantle those. Whereas Ancaps only target "the state" which they apparently largely characterize as influence on the market. So while for classical anarchists economic liberty is defined by equal ownership of the economy or at least equal access to economic resources and the ability to decide that, anarcho-capitalism is just concerned with the market and sees no problem in the economic ownership of some freedom of some leading to the total lack of economic agency and the purely theoretical economic freedom of the other.
So for all intents and purposes anarcho-capitalism is primarily concerned with capitalism and has no connection to anything related with anarchism.
If we take anarcho-socialism to mean no government and economic equality (+ collective ownership of industry), it would seem to me that this could lead to contradictions since without government intervention economic equality could not be achieved or maintained.
Afaik that's what the social ownership of the industry is meant for. Like usually the large discrepancy comes from the owner pocketing the most, whereas if everyone is an owner they can simply vote to split that more evenly. So ideally you wouldn't need "redistribution" if you're already dealing with it on the level of the production/distribution.
And the other thing is that you could argue that a huge discrepancy of stuff could be more likely to produce theft a) because there are people who don't have enough and b) because there are people who have so much that the risk/reward calculation ends up being positive. Whereas if people have roughly equal amounts of stuff, there's not much to gain.
Does socialism always presuppose the existence of a state? If so, that would make anarcho-socialism an oxymoronic, contradictory state of affairs. Are there any examples of socialist thought that rejects the state?
Technically no. Though socialism encompasses a lot of ideas and movements and some of them do propose a state, either temporarily or permanent and some don't. But primarily it's workers owning the means of production, so that could be done, at least theoretically, directly without the necessity for a state.
If libertarianism represents freedom from coercion to undertake certain actions and egalitarian socialism represents the duty to give excess wealth to those who "need it more", is libertarian socialism a contradiction of terms? Are these two principles intrinsically contradictory?
It's not about taking away excess it's about socializing production. So ideally there is no huge excess that accumulates with a small group of people. At least that is the idea.
The "giving away the excess to the poor" is usually following a different strategy. It's somewhere between a push and pull in terms of control over the economy. Like one could argue that the unequal production and redistribution is some meta level democratic ownership of the economy where the economic actors can make their own decisions but the results are shared among the constituents. Or you could argue that those with excessive stuff just rather give away stuff, then their ability to produce even more excessive stuff for themselves. So some call it socialism because of the former, while others argue it's still capitalism because of the latter, though it's somewhat of a different idea.