$\!$Hej Erik! Maybe I'm dense but I don't see what you're getting at with your first question. It's true more generally that if $Z \subset X$ is a closed subvariety of another variety, then the Zariski topology on $Z$ is the same as the restriction of the Zariski topology on $X$ to $Z$. Said differently, closed subvarieties of $Z$ are the same as intersections of closed subvarieties of $X$ with $Z$.

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 Bruhat decomposition/Schubert cell decomposition. See this question: http://mathoverflow.net/questions/151341/ and the reference to Fulton's book in Donu Arapura's answer.