left/right derivations of grammars and parse trees
A non-terminal is something with a production rule. You're correct that words aren't non-terminals. They are terminal precisely because there is no production which expands the word.
In a leftmost derivation, every step must expand the leftmost non-terminal. The leftmost non-terminal is the non-terminal which is closest to the left :-). In $\text{the N V NP}$ , the leftmost non-terminal is $\text{N}$. So if you choose a derivation step which expands $\text{NP}$, that's not a leftmost derivation.
In a parse tree, the root is whatever the root of the grammar is ($\text{S}$ in this case) and each node has as its children the symbols on the right-hand side of the production which is used the expand that particular non-terminal. Terminals are leaf nodes; they have no children.