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A geometric interpretation of the envelope of a one-parameter family of surfaces is the following (slightly adapted from Wikipedia): a point on the envelope is a point in the intersection of two "adjacent" surfaces. If you don't mind infinitesimals this could be rephrased as: $(x_0,y_0,z_0)$ is a point on the envelope, if $(x_0,y_0,z_0)\in S_{a_0}\cap S_{a_0+da}$ (where $S_a$ denotes the surface and 0-subscripts denote values of the variables $x,y,z,a$.). But this condition is equivalent to the following: The values $(x_0,y_0,z_0)$ satisfy the equation $z=w(x,y;a)$ for a certain value $a_0$, and moreover, as we change $a$ slightly, keeping $x=x_0, y=y_0$ fixed, the value of $z$ does not change to first order. This is the condition $\frac{\partial w(x,y;a)}{\partial a}=0$.

I'm not sure what it means in mathematics for something to be formal or not. So I assume you are just expressing your discomfort with differentials $dz$, $dx$ etc. Unfortunately calculus books tend to discard them as "notation without meaning". The current mainstream approach to give them meaning is to interpret them as differential forms in the sense of differential geometry. I suggest you look at any book on the topic. You would then interpret the variables $x,y,z$ as maps $x:\mathbb{R}^3\to \mathbb{R}$ etc. (And you will also be confronted with the cotangent space.)

Since they are talking about a one parameter family of flat planes through a point, it should be geometrically clear, that the intersection of two adjacent planes is a line through $(x_0,y_0,z_0)$, hence the envelope is a union of such lines and is hence a cone.