Nodes impose independent conditions

I want to compute the dimension of Severi varieties. In order to do it, I have show nodes impose independent conditions. I found this on the book "intersection theory" by Fulton. But I got some problems.

First, how does the author get (b)?

For me $$L$$ is a subspace of $$H^0(\mathbb{P}^2,\mathscr{O}_{\mathbb{P}^2}(n-3))$$. Why in (c) the author claim $$L$$ is a subspace of that $$H^0(X,N_f\otimes f^*\mathscr{O}(-3))$$?

How to show the canonical divisor is the same as that one?