Ackerman hierarchy for higher order primitive recursion in System T
(This is a very partial answer, only addressing the first point, and not the main question about the hierarchy.)

I don't have a proof to point out to you, but the idea is that a single first-order recursor on naturals such as
$$
\begin{array}{l}
rec: (nat \to nat) \to nat \to nat \to nat
\\
rec\ f\; z\; n = f(f(\ldots f(z))) \qquad \mbox{($n$ times $f$)}
\end{array}
$$

would be much less powerful than having the full (primitive) recursion scheme

$$
rec_U: (U \to U) \to U \to nat \to U
$$

since the latter roughly resembles a polymorphic term in System F, which can be used at many types $U$.

Indeed, the basic $rec$ only gives us access to the first order primitive recursion scheme, as we can find in a computability / recursion theory book.
With only that, we know functions like Ackermann's are not definable. (Well, this is not that obvious since here we also have higher-order lambdas, it's only recursion which is constrained. But the point should stand anyway, I think.)

With the $rec_U$ scheme, instead, we can choose $U = nat \to nat$, which allows us to define Ackermann's function, as Girard shows in the book mentioned by the OP. Hence, the scheme is strictly more powerful.