The way I understand these types is by thinking of what functions I can build that satisfy the signature.
Booleans
For booleans, we have the type $\Pi\alpha . \alpha \rightarrow \alpha \rightarrow \alpha$. All members of this type will take in two abstract values (that is, values of type $\alpha$, where $\alpha$ can be anything) and return an abstract value (an $\alpha$).
In System F, there are exactly two functions that match this signature: $(\lambda x . \lambda y . x)$ and $(\lambda x . \lambda y . y)$. We can name them true and false and properly claim "all booleans are either true or false".
Integers
Integers have the type $\Pi\alpha . \alpha \rightarrow (\alpha \rightarrow \alpha) \rightarrow \alpha$. All members of this type need to be functions that take in two things: an abstract value and a function on abstract values, and return an abstract value.
(Again, we don't know anything about what type $\alpha$ actually is, nor do we know what the function of type $(\alpha \rightarrow \alpha)$ does. It could be the identity, it could be the successor function, it could accept a list and return the empty list.)
One function that matches this signature is $(\lambda x . \lambda f . x)$. Another is $(\lambda x . \lambda f . f x)$. Yet another is $(\lambda x . \lambda f . f (f x))$. There are countably many of these functions, so we can put them in one-to-one correspondence with the natural numbers and name them 0, 1, 2, ... . The only difference between these and Church numerals is that you can only apply them to values $x$ and $f$ with the right type.
Lists & Trees
After integers, lists and trees are easy. The type for lists, like you said, is $\Pi \alpha . \alpha \rightarrow (U \rightarrow \alpha \rightarrow \alpha) \rightarrow \alpha$. Again, members of this type take in two arguments, an abstract value and a function that can manipulate abstract values, and returns an abstract value. The interesting part is the type $U$, which Girard uses to denote the type of elements in the list. A List Boolean has type $\Pi \alpha . \alpha \rightarrow (Boolean \rightarrow \alpha \rightarrow \alpha) \rightarrow \alpha$.
We can easily define a function nil as $(\lambda x . \lambda f . x)$. That's one way to return a list given the arguments $x$ and $f$. The only other thing we can do to return a list is to apply the function $f$; given a value $u$ of type $U$, we could make the function $(\lambda x . \lambda f . f u x)$. If we parameterize over the value $u$, we get a more familiar cons function: $(\lambda u . \lambda x . \lambda f . f u x)$. Its type is $\Pi U . \Pi \alpha . \alpha \rightarrow (U \rightarrow \alpha \rightarrow \alpha) \rightarrow \alpha$.
Trees have the type $\Pi \alpha . \alpha \rightarrow (U \rightarrow \alpha \rightarrow \alpha \rightarrow \alpha) \rightarrow \alpha$. You just add another branch!
Example: Lists in other Programming Languages
A list is one of two things:
- Empty
- Non empty, so we can think of it as one element attached to a smaller list
Let's stick to lists of integers, which Girard would describe with the type $\Pi \alpha . \alpha \rightarrow (Integer \rightarrow \alpha \rightarrow \alpha) \rightarrow \alpha$. His empty list is $\alpha$, the first argument, and his non-empty list is the second argument (which is a function expecting one element and another list).
In Java, we could represent these two alternatives by making an interface and a pair of classes.
interface IntList {}
class Empty implements IntList {}
class Cons implements IntList { int head; IntList tail; }
The Cons class has fields representing the two parts of any non-empty list.
In OCaml, things look a little more like System F.
type int_list = Empty | Cons of int * int_list
Again, there are two alternatives. A list can be Empty, or it can be a pair like Cons(2, Empty) made of an element and another list.
Girard's type is difficult to read because he expresses these ideas with one $\Pi$-type, but the idea's the same.
Traditionally $\lambda$-abstraction "associates right", while function evaluation "associates left". That is, if we see $\lambda x. \text{stuff}$, then we assume $x$ is bound for the entirety of "stuff". If we want to explicitly show where the scope of the $\lambda x$ ends, we wrap the term in parentheses.
For example:
$$(\lambda f. \lambda x . f x) w \to_\beta \lambda x . w x$$
While
$$\lambda f . ((\lambda x . f x) w) \to_\beta \lambda f . f w$$
If there aren't any parentheses, then (by convention) we assume we're in the second case. Which agrees with the example in your question. Notice in examples $1$ and $2$, the author explicitly wraps the left $\lambda$-term in parentheses so that we know to apply it.
I hope this helps ^_^
Best Answer
I don't remember ever seeing a definition of a context to include a variable. A plain variable does not have a hole, after all. But, moving on...
$F$ does not need to be well-scoped. So, for the base case where $C=y$, you can define $F$ to be $\lambda f.y$.
For the next base case, where $C[\,]=[\,]$, you can define $F$ to be the function $\lambda f. f\,\vec{x}$, where $\vec{x}$ is the $\vec{x}$ given by the statement. Then $C[M] = M$ and $F\,(\lambda\vec{x}.M) = (\lambda f. f\,\vec{x})\,(\lambda\vec{x}.M)$.
Note that in both cases, the two terms are not syntactically equal, but they are $\beta$-equivalent.