There is a lot here, so let me unpack each point one-by-one:
- Support vector machines are often used to classify points in Hilbert spaces of infinite dimension. All you need is to specify a kernel for which the induced RKHS is infinite dimensional, e.g. the Gaussian or squared exponential kernel $k(x,y)= \exp\{-\|x-y\|^2\} $. Viewing $\ell^2$ as a space of functions $u:\mathbb{N}\rightarrow\mathbb{R}$ it can be readily shown that it is a reproducing kernel Hilbert space (RKHS) with kernel $k(i,j) = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta (i.e. the function that equals $1$ only if $i=j$ and $0$ otherwise). The following is maybe a bit technical: the $L^2$ spaces are not RKHSs, since they consist of equivalence classes of functions - recall that an RKHS is a Hilbert space of functions with the added condition that convergence in norm implies pointwise convergence.
- I wouldn't say the kernel trick relies on the Moore-Aronszajn theorem. All the kernel trick relies on is the following property of a kernel $k$ and its corresponding RKHS $\mathcal{H}_k$:
$$ \langle k(x,\cdot),k(y,\cdot)\rangle_{\mathcal{H}_k} = k(x,y). $$
This property is an example of the reproducing property. Usually one would specify their kernel $k$ and then construct the feature map as $\Phi(x) = k(x,\cdot)$. By definition of the RKHS, $k(x,\cdot) \in \mathcal{H}_k$ and so $\Phi:\mathcal{X}\rightarrow\mathcal{H}_k$, where $\mathcal{X}$ is underlying set in which your data lies. Therefore, embedding your data into $\mathcal{H}_k$ using the above feature map allows one to compute inner products by just evaluating the kernel $k$. That is, we have $\langle \Phi(x_i), \Phi(x_j)\rangle_{\mathcal{H}_k} = \langle k( x_i, \cdot), k( x_j, \cdot)\rangle_{\mathcal{H}_k} = k(x_i,x_j)$ for all data points $x_i, x_j \in\mathcal{X}$. Thus, applying learning algorithms on the transformed data $\{\Phi(x_i)\}_{i=1}^n$ that rely on computations of the inner product on the data, can be easily computed by only evaluating the reproducing kernel of $\mathcal{H}_k$. The Moore-Aronszjan theorem really just tells us that, for any given a symmetric, positive-definite kernel $k$, there exists a unique corresponding RKHS. The proof is by construction.
I think my second point above answers your question on why taking inner products corresponds to evaluating the kernel. To understand the inner workings of RKHS based algorithms fully requires knowledge of some functional analysis. E.g. the Riesz-representation theorem is crucial in the definition of the kernel and in the Moore-Aronszjan theorem one constructs the RKHS by completion.
Turning now to implementation of SVMs, recall that, for data pairs $\{(x_i,y_i)\}_{i=1}^n$, we want to compute
$$ \inf_{f \in \mathcal{H}_k}\left\{ \frac{1}{n}\sum_{i=1}^n L(x_i, y_i, f(x_i)) + \lambda \| f\|_{\mathcal{H_k}}\right\},$$
where $L$ is a convex non-negative loss function and $\lambda\in\mathbb{R}_{\geq0}$ is a free parameter. The representer theorem is crucial here, since we know that the minimal $f^*$ which solves this optimisation is of the form $f^*(\cdot) = \sum_{i=1}^n \alpha_i k(x_i,\cdot),$ for some scalars $\alpha_i \in\mathbb{R}$ which, for convenience, we group together as a vector $\alpha = (\alpha_1,\ldots,\alpha_n) \in\mathbb{R}^n$. This implies that the optimisation problem can be reduced to
$$ \inf_{\alpha \in \mathbb{R}^n}\left\{ \frac{1}{n}\sum_{i=1}^n L\left(x_i, y_i, \sum_{i=1}^n \alpha_i k(x_i,\cdot)\right) + \lambda \left\| \sum_{i=1}^n \alpha_i k(x_i,\cdot)\right\|_{\mathcal{H_k}}\right\},$$
thus reducing the infinite dimensional optimisation to a finite-dimensional optimisation.
