About the Max-Cut SDP
An SDP relaxation is not an arbitrary optimization problem; rather, it consists of optimizing a linear function over semidefinite and linear constraints. One general formulation of SDP is
$$
\begin{align*}
&amp;\max \sum_{ij} c_{ij} x_{ij} \\
s.t. \;\;&amp; X \succeq 0 \\
&amp; A \mathbf{x} = \mathbf{b}
\end{align*}
$$
Here $X$ is the matrix whose entries are $x_{ij}$, and $\mathbf{x}$ is a vector whose entries are $x_{ij}$ (in some arbitrary order).

Standard accounts on semidefinite programming should explain why the SDP relaxation that you describe can indeed be formulated in this form. In short, the idea is that $x_{ij}$ is going to represent the inner product $\langle \vec{x}_i, \vec{x}_j \rangle$. The fact that $X$ is positive semidefinite is equivalent to the existence of such vectors $\vec{x}_i$. The objective function $\sum_{(i,j) \in E} \|\vec{x}_i - \vec{x}_j\|^2$ can be written as a linear combination of the $x_{ij}$, and the constraints $\|\vec{x}_i\| = 1$ can be written as linear constraints on the $x_{ij}$.

Why do we care about semidefinite programs? Because we know that they can be solved (approximately, but arbitrarily well) in polynomial time. This is not true for an arbitrary optimization problem.

There is some evidence that the Goemans–Williamson constant $0.878$ is best-possible for polynomial time algorithms. Namely, this is the case if Khot's Unique Games Conjecture holds. Under this conjecture, Raghavendra showed that SDP relaxations are optimal for a wide class of problems (constraint satisfaction problems).