Basic proofs on Weil groups

Question

I had some questions regarding the Weil group. We defined a surjective map $res: \operatorname{Gal}(L/K) \rightarrow \operatorname{Gal}(k_L/k)$, where $k_L,k$ is the residue field of $L,K$ respectively. We define the Weil group to be $W(L/K):=res^{-1}\langle Fr_{k_L/k} \rangle$, where $Fr_{k_L/k}$ is the Frobenius map $x \rightarrow x^{|k|}$. We claim that for a finite subextension $F/K$ of $L/K$:

1. $W(L / K)$ is dense in $\operatorname{Gal}(L / K)$.
2. If $F / K$ is a finite subextension, then $W(L / F)=W(L / K) \cap \operatorname{Gal}(L / F)$.
3. If $F / K$ is a finite Galois subextension, then $W(L / K) / W(L / F) \cong \operatorname{Gal}(F / K)$.

I haven't been able to follow the proofs for the same and was hoping for some references I could refer to for these proofs or if someone could provide me with proofs of the same. I do know that the following diagram holds:

but don't know how to proceed from this.