Bad News
The answer to Q3 as stated is no. Let $X$ be the Michael line, and let $Y$ be the closed subset consisting of all the rationals. Then, there is no bounded linear extension $C(Y,\mathbb{R}) \to C(X,\mathbb{R})$. A proof (I'm not sure if this is the first place where it appeared) may be found in Example 3.3 of the paper
RW Heath and DJ Lutzer, Dugundji extension theorems for linearly ordered spaces, Pacific Journal of Math, 55(2), 419-425 (1974).
On the other hand, the authors provide an extension theorem for what they call linearly ordered spaces. There is a zoo of such conditions which are related to -- but strictly stronger than -- normality under which you can find a simultaneous extension. It would be quite a painful task to try and list all of them anywhere. There was some work on the class of linearly stratifiable spaces, I think going back to work of CJR Borges in the mid-70s, but people found some gaps and some counterexamples, so I'm not sure where things stand with all that right now.
Good News
Here is one rather typical example where Dugundji extension certainly works - I'll call this assumption $N^+$. A space $X$ satisfies $N^+$ if it admits a distinguished collection of open sets $\{W(n,x) \subset X ~ \mid ~ x \in X \text{ and } n \in \omega\}$ so that
- $x \in W(n,x)$ for each $n$,
- $W(n,x) \subset W(n+1,x)$ for each $x$, and
- For any open $U \subset X$ with $x \in U$, there exists an open $V \subset X$ so that for each $y \in V$ there is some $n$ with $x \in W(n,y) \subset U$.
By the way, if $X$ is $T_1$ then this condition is equivalent to metrizability. Here is the result you want:
Theorem Let $Y$ be a closed subspace of a topological space $X$ satisfying $N^+$. If $L$ is any locally convex topological vector space, then there exists a linear map $C(Y,L) \to C(X,L)$ which produces Dugundji extensions (and in particular, satisfies the convex hull condition).
You can find the proof in a nice and short paper:
IC Starc, Concerning the Dugundji extension property, Topology and its Applications 63(2), 165–172 (1995).
