Sorry, I think I misunderstood your question; please ignore my remarks!

If one doesn't know the question, how can one give an answer? Could it be that you are actually talking about the map $x^*\mapsto s(x^*|C)$?

That the dual of $C(K,X)$ is the space of $X^*$-valued vector measures is a theorem due to Ivan Singer whose proof does not depend on the theory of tensor products; the original paper is in Russian, see Zbl 0087.31601

What is the range of the mapping $x^*\mapsto (s(.|C_n))_n$?

On the positive side, the answer to whether $\mu$ is zero is yes for Banach spaces; as proved by D. Preiss and J. Tiser, Measures in Banach spaces are determined by their values on balls. Mathematika 38, No. 2, 391-397 (1991). Zbl 0755.28006

It is known that $C(K)$, for infinite $K$, contains a copy of $c_0$, hence it does not have nontrivial type (meaning $>1$) or nontrivial cotype (meaning $<\infty$).

In the ``unbounded'' case equiintegrability does not imply relative weak compactness; consider $f_n=$ the indicator function of $[n, n+1]$ in $L_1(\mathbb{R})$.

One always has $L_q(I, Y^\bot) \subset L_p(I, Y)^\bot$. If $X$ is reflexive and separable, you might use a Hahn-Banach argument to show equality.

@Mark: Please note that $(X,\|.\|)$ is reflexive if and only if the space $(X,|.|)$ with an equivalent norm is reflexive. What August and Gerald are saying is that every separable space $(X,\|.\|)$ admits a strictly convex renorming $(X,|.|)$; applying this to $\ell_1$ or $c_0$ would give you a nonreflexive example $(X,|.|)$ where the unit ball is the convex hull of its extreme points.

Here are the references to the original papers where this result is proved (so it's true after all, but the proof is not so easy): D. O'Regan, Fixed point theorems for nonlinear operators, JMAA 202 (1996), 413-432; R. Precup, On the continuation theorem for nonexpansive maps, Studia Univ. Babes-Bolyai Math 41 (1996), 85--89. See also D. O'Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Gordon and Breach 2001, pp. 44--46. Thanks to Donal O'Regan for providing these references!

Apologies for this gap in my book that was first pointed out to me in 2015. As a result, I added a remark to the corrections page for the 7th edition, and in the 8th edition the problem appears with $X$ being a Hilbert space, where the proof seems to be clear. -- I don't remember the argument I had in mind for the case of uniformly convex Banach spaces (mea culpa!); I'll still try to hunt it down!