MCTS is not a traditional depth first minimax framework. Key concepts like alpha-beta don’t apply. Although it is proven to converge to minimax in the limit, the game trees are so large this is not relevant. You could use the network in a minimax searcher, but it’s so much slower than a conventional evaluation function it’s unlikely to be competitive.

Very roughly you can think of AlphaZero as a best-first tree search, where 'best' is some statistical estimate.

It is kind of the case, but it does not need to expand the whole node to find the maximum. It samples some children instead from a NN (the Monte Carlo aspect)