Feynman Diagrams understanding problems I'm trying to understand the basics of the formalism of Feynman diagrams describing interactions in QED and below I present two examples where I still don’t understand the logic behind them:
Image 1 (found here,
picture 6.8):

Image 2 (found here;
see first image in Jay Wacker's answer):

I learned that both degrees of freedom in the 2D Feynman diagram
(aka the horizontal and the vertical axis) represent the space axis and time axis or vice verse; see e.g. here:
https://en.wikipedia.org/wiki/Feynman_diagram#Description
So the $x$-axis is time and $y$-axis is space or vice versa, but they are fixed.
But in both cases I not understand the meaning of vertical trajectory paths  or more precisely the lines which are parallel to space axis.  What does it mean if in a Feynman diagram a trajectory of real  (not virtual) particle moves "parallel"
to space axis like in both examples above? Parallelity to space axis suggests that the whole trajectory is passed at the same time. Does it make sense?
For example in Image 2 (from Jay Wecker's image) we start with two gluons and every gluon produces a pair of quarks and so on. Then seemingly the horizontal  axis is then the time axis, so the process evolves temporally from left to right. But then I don’t understand what does the  vertical trajectory on the left of one of the two quarks $t$ mean? The whole trajectory is passed at same time?
Image 1 contains the same problems. The photons generate a $e^- e^+$ pair. But the behavior of the trajectories of the particles I don’t understand. The square has two $e^-$ and two $e^+$ particles.
If we choose one axis as space axis then the whole trajectory of one pair  takes at same time. This doesn’t make any sense.
Could anybody explain to me the errors in my thinking?
 A: There are theorems of QED which say that only the topology of the diagram matters. This means that the angle of the line has no meaning, as we integrate over all possible vertex positions anyway. So a given diagram represents a family of processes. Also, internal lines in a Feynman diagram do not have to be on-shell- meaning that they may have spacelike momentum.
A: This is a comment, to help in understanding Feynman diagrams.

The American theoretical physicist Richard Feynman first introduced his diagrams in the late 1940s as a bookkeeping device for simplifying lengthy calculations in one area of physics—quantum electrodynamics, or QED, the quantum-mechanical description of electromagnetic forces.

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Feynman introduced his novel diagrams in a private, invitation-only meeting at the Pocono Manor Inn in rural Pennsylvania during the spring of 1948. Twenty eight theorists had gathered at the inn for several days of intense discussions. Most of the young theorists were preoccupied with the problems of QED. And those problems were, in the understated language of physics, nontrivial.

....

Although the full calculations extended in principle to include an infinite number of separate contributions, in practice any given calculation could be truncated after only a few terms. This was known as a perturbative calculation: Theorists could approximate the full answer by keeping only those few terms that made the largest contribution, since all of the additional terms were expected to contribute numerically insignificant corrections.


Deceptively simple in the abstract, this scheme was extraordinarily difficult in practice. One of Heisenberg’s graduate students had braved an e4 calculation in the mid-1930s—just tracking the first round of correction terms and ignoring all others—and quickly found himself swimming in hundreds of distinct terms. Individual contributions to the overall calculation stretched over four or five lines of algebra. It was all too easy to conflate or, worse, to omit terms within the algebraic morass.

....

In his Pocono Manor Inn talk, Feynman told his fellow theorists that his diagrams offered new promise for helping them march through the thickets of QED calculations. As one of his first examples, he considered the problem of electron-electron scattering.

Please read the article for the history of Feynman diagram use, how it managed to pictorially represent complicated QED integrals.
In answer to your questions, the original integrals represented finally by the diagrams, did not give rise to such questions,  time and space were variables in a complicated integration  . The only fixed  real numbers were the input four momenta , and the output of the calculation giving the crossection value.
As this is a comment, I have a Feynman story that I heard from Feynman himself when he participated in a workshop in Crete in 1980. (  I have said it in different words here too, as an answer to a soft question)
He said that during the Manhattan project
(The Manhattan Project was a research and development undertaking during World War II that produced the first nuclear weapons) the theorists gathered were given a specific problem for calculating and in a week each came with his calculations and presented to  all, a sort of multiple check of calculations needed for the project, mainly crossections of interactions. These calculations were carried out in the perturbative method, described in the paper referenced above.
Then, and he said it so vividly that it is if I saw him, approximately :"one night, while calculating  I was lying on the bed resting my feet on the wall when I saw how to do this fast, all in a flash"  . He then did the calculation that would have taken him a week in a day and a half. When he went to the weekly gathering, he saw it was confirmed by the laborious calculations the other scientists (see the bottom of the wiki article for scientists participating) had carried out.
He tried this for some weeks, and then he started kidding the rest of the  group. He would tell them the final calculation a few days ahead of the weekly meeting, and it would be confirmed!!
At the time he said this, I thought he meant he visualized the Feynman diagrams and used them in his calculations.
A: 
Could anybody explain to me the errors in my thinking?

Your fundamental error is thinking that the lines in a Feynman diagram are actual trajectories. Quantum particles don’t have trajectories.
Furthermore, virtual “particles” aren’t real particles; they don’t even obey basic relations between the energy, momentum, and mass of a real particle, such as $E^2-\mathbf{p}^2=m^2$.
A Feynman diagram is simply a pictorial representation of mathematical term in a perturbative expansion of a transition amplitude. It isn’t a picture of things moving around.
