The point of defining the 1PI diagrams is that calculating all the diagrams (including the reducible ones) is redundant. Say you already calculated the lowest order for the self energy, the diagram with an electron loop. What if you have an electron loop and then another electron loop (so a bit like your second picture)? Is that any harder? No, it's just the value of one electron loop squared. So if we just calculate the 1PI diagrams we can get all the diagrams with very little extra work.
All possible 1PI insertions just means that, that inside of the shaded circle you can put anything as long as it connects with the external lines and it's 1PI. The pictorial intuition is a good intuition, because, again, the point of the 1PI diagrams is to simplify. If a diagram can be separated in half by a single cut, then it's just the product of two simpler diagrams.
As for the propagator, recall that earlier in the book you calculated the two point function for the scalar field and it turned out to be the sum of all diagrams with two external points at fixed positions $x$ and $y$. The Fourier transform is just the momentum space version of that. And the sum of all possible diagrams with two external photons is what P&S draw in the second picture, by definition of 1PI: if some diagram is not 1PI, it can be separated into two 1PI pieces.
In regards to your edit: what the book claims is only true if you include all diagrams at a given order. In your example, you've missed that you should also have a $I'-II$ and a $I-II'$ diagram. The sum of all four will give what you want.