I guess I'll say something about question 1 beyond my comment above.

Below are a few points I took away from a graduate class on generating functions taught by Stanley and from an undergraduate class taught by Nantel Bergeron, the latter of which discussed André Joyal's theory of species for a few weeks. Clearly there are other MO users with more expertise on this than me, and they should feel free to edit my post if they are so inclined.

Exponential generating functions $\sum_{n\ge 0} a_n \frac{x^n}{n!}$ are handy for enumeration problems involving labeled objects, while ordinary generating functions $\sum_{n\ge 0} b_n x^n$ are more convenient for unlabeled objects. As an example, exponential generating functions work well for problems involving set partitions, i.e. ways of splitting a set $\{ 1,\dots ,n \} $ of distinguishable objects into blocks, while ordinary generating functions tend to work better for enumeration problems regarding integers partitions.

The operation of taking the derivative of a generating function $\sum_{n\ge 0} a_n \frac{x^n}{n!}$ and then multiplying by $x$ corresponds naturally to switching from having coefficients $a_n$ counting unmarked objects of size $n$ to a new generating function whose corresponding coefficient $na_n$ counts marked objects of size $n$. For instance, if $a_n$ counted committees of size $n$, then $na_n$ would count committees with a choice of chairperson.

Multiplying generating functions $\sum_{r\ge 0}a_r x^r$ and $\sum_{s\ge 0}b_s x^s$ corresponds to switching from having coefficients $a_r,b_s$ respectively counting objects of size $r,s$ in sets $A$ and $B$ to now letting the coefficient of $x^{r+s}$ in the product count the ordered pairs in $A\times B$ whose orders sum to $r+s$. While I wrote this for ordinary generating functions, there is also an analogous version for exponential generating functions where binomial coefficients do just what you would hope.

A consequence of 3 is that exponentiating an exponential generating function $F(x)$ to get $e^{F(x)}$ corresponds to switching from counting objects of size $n$ to counting collections of these objects where the orders of the elements in the collection sum to $n$. A concrete example is where the cofficients in $F(x)$ count permutations comprised of a single cycle of size $n$, whereas the coefficients in $e^{F(x)}$ count all permutations of size $n$ (having any number of cycles).

The various books, etc., written in the 40 years since that quote do quite a lot to systematize things. For instance, there is Enumerative Combinatorics, Volume 2, by Richard Stanley.

Regarding species, one thing I liked was that there were very nice diagrams people drew to depict operations such as above which were actually helpful for deducing combinatorial identities (as I recall). One or two of these pictures appear at: http://en.wikipedia.org/wiki/Combinatorial_species

Our discussion of species in the class of Bergeron was prefaced by a quote from our professor which I liked very much: that undergraduate classes deal primarily with categories while graduate classes deal mainly with functors. I think he said this to help prepare us for facing a functor.