On spacetime diagrams cosmologists represent our past light-cone as a two-dimensional surface extending back in time, on which are situated all light-emitting events which we can observe today. This is especially clearly shown when cosmic comoving coordinates are used. But shouldn't the past light-cone also have some (causal) extension in the 3rd spatial dimension and how large could this "thickness" be?
You're absolutely right, but how would you propose drawing that on a piece of paper? Spacetime is four-dimensional with lorentzian signature. Paper is two-dimensional with euclidean signature.
Remember the equivalence principle: in any sufficiently small neighborhood, spacetime is approximately flat. Your intuition about light-cones in flat spacetime is a good approximation in any region that is small compared to the cosmological scale (ignoring tiny blemishes like galaxies, as cosmologists usually do). Large-scale curvature distorts the light-cones, but it doesn't flatten them. To give a more specific answer may require knowing exactly what pictures you're looking at, but the bottom line is that you're right. It's just hard to draw.
If a picture is worth a thousand words, then an equation is worth a thousand pictures. Math communicates concepts that pictures can't even begin to touch. A good picture can help us ease into the math, and it can help us check our understanding of the math, but only in very simple cases can a picture actually communicate everything that matters. Cosmology is not one of those cases.