Why is atomic orbital one electron wave function? Why distance from centre is proportional to angular wave function?
OK, let's address these one by one.


Yes, it should be a two-electron wave function, i.e., the one that depends on 6 spatial coordinates. Just imagine the amount of trouble required to simply write it down, let alone produce a meaningful plot or do anything else with it. So we go with two one-electron functions instead (see Hartree-Fock method). That's an approximation, of course, but it would suffice for now.
Yes, in angular wave function plots, it is the numerical value of $\Theta\Phi$ that is actually plotted by means of the distance from the origin. There is no physical reasons behind this. It is not a physics question at all. It is a question about drawings made by humans on pieces of paper. We may plot whatever we like against whatever else we like. If you find the type of plot which gives you better intuition of what the spherical harmonics look like, you are welcome to use those. (Hint: people have been using these for quite a while now.)
The "not the distance" part of the quote is cryptic to me. I can only guess it is a botched attempt to point out that the plot is not related at all to the radial part of the wave function, and hence to the distance from the electron to the origin.
Yes, that would require a four-dimensional graph, which we can't do on paper. Wait, we can't even do a three-dimensional graph on paper. That's why we resort to cross-sections and other weird plots in the first place.


So it goes.