The Jacobian criteria is all what you need. A point x of the quintic is singular if and only if all the partial derivatives of the defining equation of the quintic vanish at x. First write the partial derivatives with respect to variables different from $x_0$ and $x_1$. As $f$ and $g$ are generic, the vanishing of these derivatives imply $x_0 = x_1 =0$. Then compute the partial derivatives with respect to $x_0$, $x_1$ and use the fact that we already know that $x_0 = x_1=0$ to conclude that $f(x)=g(x)=0$ for a singular point. This shows that a singular point $x$ necessarely satisfies $x_0 = x_1 = f(x)=g(x)=0$. As conversely any $x$ satisfying these conditions is on $X$, the solutions to these conditions are exactly the singular points of $X$.

As $f$ and $g$ are generic of degree 4, the equations $f=g=0$ define a surface in $\mathbb{P}^4$ of degree 4.4=16 and so its intersection with the plane $x_0=x_1=0$ is made of 16 points.

As $f$ and $g$ are generic, they have some non-trivial linear term around each singular point, in particular the defining equation of the quintic has a leading non-trivial quadratic form around each singular point, i.e. each singular point is a node.