Forgive me but every time I hear this it sounds completely abstract. What you mean "it abolishes the mode of production..."?

Marx explained the progression of history by saying that it went through a series of different 'modes of production'. Essentially they're different methods of organisation, with different class relations, and different factors of production. The capitalist mode of production is characterised by wage labour (rather than slave or peasant labour), private property and the extraction of surplus value, among other things.

Communism is the movement of the working class to seize the means of production and control them themselves, through democracy. That way we'll end class society, and through the implementation of a planned economy we can end horrific economic inequality.

I get that it's fairly abstract, but Marxism typically is.

It's so abstract, I could (and will) argue that ANCAPISM abolishes a mode of production which is used to divide the classes, namely the wielding of statist guns by giant corporations.

Well it obviously doesn't abolish class in the Marxist sense, because it maintains capitalism and thus maintains a division between those who own the means of production and those who labour in them. Moreover it doesn't abolish the power of corporations or even abolish the state because you just want to create a load of competing, private states. All of the problems with the state we have now will be maintained but without even the scrap of democracy that we currently have. The guns will still exist, and they'll still be trained upon the working class.

There would still be wealth differences, but they would be earned more and stolen less compared to the status quo.

lol, no. All wealth earned by capitalists is stolen from what their employees create. That will stay exactly the same in ancapistan.