I'm searching for a recommendable reference dealing with theory of non-Archimedean local fields where I can find proofs of the following claims about finite extensions $L/K$ of non-Archimedean local fields with finite residue fields $l / k$. I'm pretty sure that this request might not have a research level but up to now I haven't obtained a satisfying answer asking the same at MSE.

Firstly, we use notations: We denote by $q \in O_K$ a uniformizer of $O_K$ and $\pi \in O_L$ a uniformizer of $O_L$. Then $n= [L:K]=e \cdot f$ with $f= [l:k]$ and $(q)O_L= (\pi)^eO_L$.

Now I'm looking for rigorous proofs of the following statements:

(i) Case $L/K$ is Galois & unramified ($\Rightarrow$ $e=1$ & $[L:K]= [l:k]$) and $p$ the characteristic of finite field $k$:

The claim is $G= \operatorname{Gal}(L/K)= \operatorname{Gal}(l/k)=g$. I'm looking for a proof making an explicit construction of the "lift" $\operatorname{Gal}(l/k) \to \operatorname{Gal}(L/K)$, i.e. if we have a $k$-automorphism of $l= k(a)= k[X]/ (X^f-1)$, how can it be uniquely lifted to a $K$-automorphism of $L$?

(ii) If $L/K$ is unramified and $K= \mathbb{Q}_p$, then $L$ is cyclotomic: $L= \mathbb{Q}_p(\zeta_n)$, for $\zeta_n$ an appropriate root of unity.

(iii)(1) Case $L/K$ totally ramified with $n =e$ coprime to $p=\operatorname{char}(k)$: there exists $b \in K$ with $L= K(\sqrt[e]{b})$.

How can this $b$ be constructed? Can the $b$ be chosen as a uniformizer: i.e. $b= uq$ with $u \in O_K^\times$?

(2) If moreover $p \mid \lvert k \rvert -1$, why is $L/K$ cyclic & Galois extension?