For any cover $U^{\bullet}$ and any sheaf $F$, there exists a Cech complex $j_!^{\bullet} F\rightarrow j_!F$. Now, you just take the hypercover $U^{\bullet}$ and $F=\mathbb{Z}$. Maybe you can see section 2.8 of Sheaves on manifold of Kashiwara-Schpira. We don't need hyper condition here.

To incorporate with $C$, you need to notice that $RHom(j_!\mathbb{Z},C)\simeq R\Gamma(U,C) \simeq \Gamma(U,C)=C(U)$ for all open sets $U$, where the second isomorphism follows K-injectivity. K-Injectivity also tells you the resulting sequence is actually resolution.

For any cover $U^{\bullet}$ and any sheaf $F$, there exists a Čech complex $j_!^{\bullet} F\rightarrow j_!F$. Now, you just take the hypercover $U^{\bullet}$ and $F=\mathbb{Z}$. Maybe you can see section 2.8 of Sheaves on manifolds of Kashiwara–Schapira. We don't need hyper condition here.

To incorporate with $C$, you need to notice that $R{\operatorname{Hom}}(j_!\mathbb{Z},C)\simeq R\Gamma(U,C) \simeq \Gamma(U,C)=C(U)$ for all open sets $U$, where the second isomorphism follows from K-injectivity. K-Injectivity also tells you the resulting sequence is actually a resolution.