$\!$Hej Erik! Your first question asks why an open Schubert cell has the same closure both in the Zariski and in the classical topology. This is the same as asking why the closure of a cell is a closed subvariety. But each cell is defined by a collection of polynomial equations ($f = 0$) and inequations ($f \neq 0$), and it's easy to check that the closure in the classical topology is obtained by just discarding the inequations. Then it's clear that the closure is a subvariety.
For your second question, this follows because Grassmannians admit an algebraic cell decomposition (i.e. they are the disjoint union of locally closed subvarieties isomorphic to some $\mathbf C^d$), the Schubert cell decomposition. See this question: For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism? and the reference to Fulton's book in Donu Arapura's answer.
