Exercise 11 of section 3.2.2(The art of computer programming)
We have $z^\lambda$ has the form of
  $$f(z)u(z) + p^ev(z) + 1$$
  where $p^{e+1}\mid f(z)u(z),f(z)\mid p^ev(z)$ and $v(z)\not\equiv 0\pmod p\ $  (*)  


You misunderstood the definition of $a(z)\equiv b(z)\pmod {f(z)\text { and }m}$. If $z^\lambda \equiv 1 \pmod{ f(z)\text{ and }p^{e}}$, then $$z^\lambda=1 + f(z)u(z) + p^ev(z)$$
for some $u(z), v(z)$. There is no further requirement on $u(z)$ and $v(z)$. It is incorrect to deduce $p^{e+1}\mid f(z)u(z)$. It is incorrect to deduce $f(z)\mid p^ev(z)$.

However, if we also have $z^\lambda \not\equiv 1 \pmod{ f(z)\text{ and }p^{e+1}}$, then we must have $v(z) \not\equiv 1 \pmod{ f(z)\text{ and }p}.$




  Why is the above argument necessary?


Now that you can understand the notation correctly, you can figure it out by yourself.




  Is the Gauss's lemma 4.6.1G the one in Wikipedia?


Yes, lemma 4.6.1G in the book is the same as the Gauss's lemma on Wikipedia.