I've taken a look at the book and this chapter is introductory and somewhat informal, so I imagine the authors are more specific about what they mean by a white noise in space or time and what they mean by the S(P)DE in your question in later chapters. Nevertheless, I have addressed aspects of your question below.
A discussion of Definitions 2, 3 and 5 are contained in an answer of mine to a similar question here. Everything in that answer is real-valued (which hopefully doesn't make too much of a difference) and indexed by a single real variable (or more precisely a test function of a single real variable); this can make a significant difference depending on what you want to know.
Definition 2
The random distribution that acts on $\phi$ via $(W, \phi) = \int \phi(t) B_t dt$ is just the Brownian motion $B$ (i.e. we can identify the function $B$ with the distribution $W$).
Your definition of $W'$ is then how I define white noise (denoted $X$) in the answer linked to above: white noise $X$ is defined as the random distribution that acts on a test function $\phi$ by $(X, \phi) = -\int_0^\infty B(t) f'(t) dt$. In the parlance of the book you cite, this is a white noise in time (time is the only variable in that answer). However, you can generalize this definition to white noise in space and time (see the discussion of Definition 3 below).
Definition 3
Here $W$ is your white noise (not $W'$ as in Definition 2).
To link this to definition 2, set $d = 0$ (so there is no spatial component to the domain of $\phi$). With $X$ defined as above, $(X_\phi := (X, \phi) : \phi \in C^\infty([0, \infty))$ is a centered Gaussian process with covariance $E(W_\phi W_\psi) = (\phi, \psi)_{L^2}$ (by the Ito isometry). The definition you have stated is a generalization to the case where the process is indexed by space and time (more precisely by test functions of space and time).
Definition 5
Your definition of $W$ is the same (by stochastic integration by parts) as the definition of $X$ above. Thus, $W$ here is once again white noise ($W'$ is then the distributional derivative of white noise).
Definition 1
In this definition, while the realization of the process you get in this way depends on the choice of basis, its (probability) distribution is independent of basis. You can think of a white noise as any process with this distribution.
This definition must be understood in the sense of distributions (now referring to Schwartz distributions) as white-noise is not defined pointwise (so $W_t$ is meaningless). A more precise definition is that $W$ acts on a test function $\phi$ by $W_\phi := (W, \phi) = \sum_{i=1}^\infty \xi_i (\phi, \phi_i)$. Now you can check that $W_\phi$ has mean $0$ and that
\begin{equation}
E(W_\phi W_\psi) = E\sum_{i=1}^\infty \xi_i \xi_j (\phi, \phi_i) (\psi, \phi_j) = \sum_{i=1}^\infty (\phi, \phi_i) (\psi, \phi_j) = (\phi, \psi)_{L^2}.
\end{equation}
Thus, the only thing to check to see that $W$ has the same distribution as the processes above is that it is Gaussian.
Your question is ill-formulated as is. To have a well defined measure, you need to give a definition of your bilinear form $\langle \cdot,\cdot\rangle:\mathcal{S}'\times\mathcal{S}'\rightarrow\mathbb{R}$. For arbitrary temperate distributions $F$, $G$, what you wanted to define is, I presume,
$$
\langle F,G\rangle=(F\otimes G)(\phi)
$$
where $\phi(x,y)=v(x-y)$. The problem is that even if $v$ is in $\mathcal{S}$, $\phi$ is not a Schwartz function. In particular if $F=G=1$ (the constant function equal to 1), the bilinear form does not make sense, because you have decay in the $x-y$ direction but no decay in the $x+y$ direction.
Best Answer
Simon mentions this in passing, so without details. Justifying this needs some extra assumptions on $V$ and some extra work. A case where this can be done easily is if $$ V(x-y)=\iint \rho(x-u)W(u-v)\rho(y-v)\ du\ dv $$ where the mollifier $\rho$ is in $\mathcal{S}$ and where $W$ is in $\mathcal{S}'$ and satisfies the positivity condition $$ \iint f(u)W(u-v)f(v)\ du\ dv\ \ge 0 $$ for all $f\in\mathcal{S}$. Then you can use the Gaussian measure on $\mathcal{S}'$ with covariance $W$ intead of $V$ to prove the identity. You just need to evaluate the characteristic function on the test function $$ \sum_{i=1}^{n}a_i\rho(x-x_i)\ . $$