Hello, I do not speak English, so why I write is what the translator of Google translate in my place, I do not think this means an insurmountable disadvantage because I can read to you without drawbacks, however Anador think my message in language Original (Spanish) case.
Welcome to the board. Hopefully the translation should not cause to many problems, but translation software is not perfect, so misunderstandings may arise.
The first thing I want to point out is that it is too coincidental that the only mission went wrong, the only one with such bad luck has been the No. 13 ... which could be a coincidence but it also decided that NASA took off at exactly 13 hours and 13 minutes ... 
WOW !!! one would think the script for this movie had been previously written.
Coincidences happen. In the case of the Apollo 13 incident, too many small and unrelated things came together for there to be any great significance. The launch time was 13.13 at Mission control, 14.13 at the launch site, and 19.13 GMT (or UTC). This timing was dictaed by the planned landing site.
The second thing that has struck me is these strange images where we see the separate command module service module in lunar orbit (?) Why this maneuver would do if NASA says the CSM completely returned to Earth?
They are not in lunar orbit, only flying round the Moon once to return to Earth. The pictures are taken through the docking window at the top of the LM (visible on the video you linked to at 20:00), and from that angle the conical command module is clearly visible, but the cylindrical service module is hidden behind it. It's still there, just not visible in the picture due to perspective.
The third question comes from this video, and section runs from 19:23 to 20.11 minutes, we see Fred Haise being filmed and then the camera goes to LM where the other two atronautas appear ... who filmed this scene if only were three on board?
Haise was holding the camera. Notice he only appears briefly at the edge of the frame, we never see all of him, and he has one arm extended out of shot. Movie camera selfie, and probably unintentional.
Fourth I have deep doubts about the flight plan provisions, as NASA crew used the lander as a lifeboat, and here are two issues to resolve, the first is how could the small motor LM to get the drive and autonomy necessary if (as the very NASA) Armstrong barely six seconds of fuel left over for Apollo 11 landing on the moon, with the aggravating circumstance that now had to carry the dead weight of nearly 30 tons of useless weight of CS.
It did not need to. It only had to adjust the course of the spacecraft that was already moving at several thousand mph such that it would swing around the moon and return to Earth.
And this is where we find the pretext advanced by NASA (?) They did so to protect the heat shield of the capsule and this does sound like nonsense, why would you look both toughest part of the whole Saturn rocket? if the fragile construction of the LM had no objection to travel through empty space, it is not then understand much care for a shield designed to survive the terrible demands imposed reentry into the atmosphere.
The heat shield is not the toughest part of the craft except in the situation of re-entry. It was designed to char and burn away, thus taking the heat of re-entry with it and away from the command module. However, if it were exposed to the vacuum of space, which is a very effective heat sink, the cooling process might crack the heat shield, or it might impact some of the debris from the oxygen tank explosion, or uneven heating of the spacecraft by the Sun might cause it to crack. That would make it useless. Better to protect it from all those things and make sure it would work when it was needed.
