I think that question Q3 has nothing to do with Banach space valued functions: If $Y$ is a closed subset of a compact space $X$ such that there is a continuous linear extension operator $F:C(Y)\to C(X)$ then one can use the injective tensor product to get an extension operator $$F\otimes id_E: C(Y,E) \cong C(Y)\hat\otimes_\varepsilon E \to C(X)\hat\otimes_\varepsilon E \cong C(X,E)$$ even for every complete locally convex space $E$. The isomorphism $C(X)\hat\otimes_\varepsilon E\cong C(X,E)$ is desribed, e.g., in Jarchow's book Locally Convex Spaces, chapter 16.

I would be very surprised if it would be unknown whether the restriction operator $C(X)\to C(Y)$ always has a continuous linear right inverse.

I think that question Q3 has nothing to do with Banach space valued functions: If $Y$ is a closed subset of a compact space $X$ such that there is a continuous linear extension operator $F:C(Y)\to C(X)$ then one can use the injective tensor product to get an extension operator $$F\otimes id_E: C(Y,E) \cong C(Y)\hat\otimes_\varepsilon E \to C(X)\hat\otimes_\varepsilon E \cong C(X,E)$$ even for every complete locally convex space $E$. The isomorphism $C(X)\hat\otimes_\varepsilon E\cong C(X,E)$ is desribed, e.g., in Jarchow's book Locally Convex Spaces, chapter 16.

I would be very surprised if it would be unknown whether the restriction operator $C(X)\to C(Y)$ always has a continuous linear right inverse.

Edit. I just learned from Tomasz Kania's answer to this question Nonseparable counterexamples in analysis that an example is $X=\beta\mathbb N$, the Stone-Cech compactification of $\mathbb N$, and $Y=\beta\mathbb N\setminus\mathbb N$: If there were an extension operator $C(Y)\to C(X)$ then the kernel of the restriction operator would be complemented in $C(X)=C(\beta\mathbb N)=\ell^\infty(\mathbb N)$ but for the remainder $Y$ this kernel is $c_0$ which is not complemented (the theorem of Phillips from 1940).