This is a series of questions in chronological order. I am lately trying to understand Okounkov's Random surfaces enumerating algebraic curves. So he mentions something about virtual fundamental class.

1. Can someone give a little intuition or exposition about what this is in the context of section 2.1.3 of the paper?
2. Another question is how does one define $H_2(X)$ in section 2.2.2 (I apologize for the elementary nature of the question in the eyes of algebraic geometers, but I am only familiar with $H_k$ for topological spaces); I presume one needs some sort of etale cohomology? And what does he mean by "the hyperplane class induced from the ambient $\mathbb P^N$" in the same section?
3. In section 3.1.3, the author says in the second paragraph that "the total width of these infinite rows and columns(2, in this example)", why is it $2$? Also he says hte constant term $\chi$(=9 here), immediately after. Why is that $9$? I presume he is talking about figure 2, but I couldn't see the $2$ or the $9$ in the diagram.

That's it for now, probably will have more questions later. But as usual help is greatly appreciated!