How to find $\phi$, that denotes the correlation of signals among informed traders?
A degenerate joint normal is distribution is one in which you cannot find a PDF for the distribution. They assume you can. (The covariance matrix is invertible).

Let $f(s_1,s_2\dots,s_n,v)$ be the distribution.
If I was to exchange $s_1$ for $s_2$, $f(s_2,s_1,\dots,s_n,v) = f(s_1,s_2\dots,s_n,v)$, the distribution does not change. And you can exchange as many indices (signals) as you'd like.

The assumption that the distribution is jointly normal implies $\tilde{v}$ is an affine function of $\tilde{s}_i$, which then implies the sum of signals.

Regress $\tilde{v}$ on $\tilde{s}_i$ and calculate $R^2$ (the Coefficient of determination).

This is something I hope someone else can answer.