Not really a fan of this question in how it is posed. I'm assuming LS = libertarian socialism. I don't identify as a libertarian socialist, though I suppose I could still be lumped under this term as an anarchist.
1. Market anarchism, typically collectivism, mutualism and some forms of individualism are not necesssarily against markets, exchange or the use of money in paper form. Some may have a more nuanced argument on the type of money to be used, but generally market anarchists tend to view anarchist communism as one choice of market anarchists could have. I don't agree with this view and find markets to be part of creating systems of domination.
2. All anarchists while under capitalism can, if they own property, turn that property to common use to "benefit society" or they may use their property in a way that creates a benefit to society. Market anarchists tend to have amended views on property that vary based on the anarchist, but typically the different with pro-capitalists is based on possession. What a person possesses is theirs and they can turn a use towards it, but what isn't part of a person's possessions may be used by others seems to be a typical position for some market anarchists, though others might have a stronger sense of property that mirrors the current systems views.
3. Collusion between the church and the state is common enough to see the church as an institution of domination, which makes it a target. There may be some that view a secular society's relationship with religion to be less dominating, but the reasons for the anarchist communists of Spain, in that period, to attack the church, I accept and don't lose any sleep over. Differences of opinion will always occur and most anarchists are willing to maintain relationships that exclude domination from their lives.
4. I'm not certain of what kind of answer you are looking for on this point. There is not a single definition to any word, especially key words that are of principle to how anarchists view the world, like authority. Libertarian socialism is broad enough to include non-anarchists, like libertarian Marxists or anti-state liberal activists, in some interpretations. In other interpretations libertarian socialism = anarchism, so the noted examples wouldn't be considered part of this tradition.
As noted above, market anarchists exist, though to other non-market anarchists, they find their interpretation to be splitting hairs at best and at worst, not really anarchists and in favor of an ideal capitalism rather than being against capitalism.