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The Fermi interaction is the low-energy four-fermion approximation to the weak charged current. Integrating out a W boson gives its coefficient $G_F/\sqrt{2}=g^2/(8M_W^2)$ at tree level.

Allowed. A single $t$-channel W boson mediates the weak charged current transitions $u\to d$ and $\bar{c}\to \bar{s}$. The amplitude is proportional to the Cabibbo-Kobayashi-Maskawa matrix product $V_{ud}{V}_{cs}^{\ast }$.

Allowed. There are two tree-level Feynman diagrams: $t$-channel Z boson exchange through the weak neutral current, and $s$-channel $W^{+}$ formation through the weak charged current.