Hypothetically, the moment that the state doesn't exist, private property would be unenforceable/meaningless. How the state might be abolished is another question. While marxist-leninists might opt for a "trust me bro the state will dissolve once we reach true communism" strategy, anarchists again would not. Obviously it's a tough question and an even tougher endeavour but it's not something anarchists don't have answers for.

Anarchist praxis is more grassroots focused, because obviously anarchy/anarcho-communism can't happen overnight, and requires a profound change in social values. If you choose to believe that it's impossible, I can't blame you, but I would like to ask why.

Also you use private property and "property" interchangeably, but there's a difference between personal property and private property. Abolishing private property isn't about taking away your personal house and toothbrush, it's about abolishing landlords, factory owners, and other "private" ownership of public/communal spaces.

Regarding unions, it depends whether or not they've been structured specifically without hierarchy in mind. In any case the point is that anarchist praxis is diverse and decentralised.