PTAS definition vs. FPTAS
Let me answer your questions in order:


By definition, a problem has an FPTAS if there is an algorithm which on instances of length $n$ gives an $1+\epsilon$-approximation and runs in time polynomial in $n$ and $1/\epsilon$, that is $O((n/\epsilon)^C)$ for some constant $C \geq 0$. A running time of $2^{1/\epsilon}$ doesn't belong to $O((n/\epsilon)^C)$ for any $C$.  An algorithm whose running time is $O((n/\epsilon)^C)$ is better than an algorithm whose running time is only guaranteed to be $O(n^C e^{D/\epsilon})$, since the dependency on $\epsilon$ is better for the first algorithm. Furthermore, for any $E$, we can find an $1+1/n^E$-approximation in polynomial time using the first algorithm but not using the second (at least not with the given guarantee).
A problem in which an $1+\epsilon$-approximation can be found in time $(n+1/\epsilon)^3$ is definitely in PTAS, since for every $\epsilon$ this is $O(n^3)$ (exercise) and so polynomial in $n$.
What the authors meant here is that since an NP-hard optimization problem can't be solved exactly in polynomial time, the best we can hope for is for it to be $\epsilon$-approximable in polynomial time, and furthermore with a good dependence on $\epsilon$. Among the common complexity classes, FPTAS gives the strongest guarantee on the dependence on $\epsilon$. But in practice we sometimes get an even better guarantee: the running time is polynomial in $n$ and in $\log(1/\epsilon)$. So it's not strictly true that FPTAS is the strongest possible result; it's only the strongest possible result among the options PTAS,FPTAS,P. If we made up a new class LPTAS (corresponding to time polynomial in $n$ and in $\log(1/\epsilon)$), then that would be a stronger guarantee.
Given an NP-hard optimization problem, it can't be solved exactly in polynomial time; the best one can hope for is to approximate it efficiently. Some problems are NP-hard to approximate to some constant $C&gt;1$. For others, it is possible to approximate the problem arbitrarily well in polynomial time, and these problems have a PTAS and so belong to the class PTAS. It might be that a $1+\epsilon$-approximation takes time proportional to $e^{1/\epsilon}$, and so we can only apply this efficiently for constant $\epsilon$. If the problem has an FPTAS (and so belongs to the class FPTAS), then we know that the dependency on $\epsilon$ is only polynomial, and so we can approximate efficiently to within $1+1/n^C$ for any $C$.