Socialism by definition is not anti establishment. It's anti current establishment but the philosophy is geared towards a bigger government "establishment".
No, it is not. Socialism does not mean statism. Please feel free to read up on libertarian socialism, Anarcho syndicalism or even eco socialism. The Greens/EFA in europe lean towards libertarian socialism.
I'm pretty sure that's the difference between anti establishing and anti government. the "establishment" is the established power structure. technically fascists ARE anti establishment in most places. and that's a very good thing.
Republicans are going mask off, Italy is run by fascists again, AfD is at 20%, Putin is invading neighbors and bankrolling white supremacists world wide, Israel's engaging in open genocide, their neighbors wish they could do the same, the second largest party in Sweden was founded by Neo-Nazis...
hmm, that feels a bit like splitting hairs. at that point we have to argue whether the establishment is capitalism, or the government. and that just seems like a pedantic argument about semantics that will accomplish nothing. point being, most fascists want to overthrow their current government.