Avoir. Porquoi ne c'est par le verbe avoir in the infinitive aprés the mot 'aprés?

Not totally sure I understand your question but here goes! In French, the way to say "after doing something ..." is "après avoir fait quelque chose ..." when the subject of both actions is the same, so in this case "after having tried in vain to unlock the car ... I called a taxi" translates as "après avoir en vain essayé d'ouvrir la voiture ... j'ai appelé un taxi" because in this case both actions are being done by ME. But later in the exercise the subject in the "after" sentence is different - "after HE pushed me from the taxi, it was ME that succeeded in staggering to the door" so that's why no "avoir" in that case (if that is your question) and it is simply "Après qu'il m'a poussé ... j'ai réussi ..."

If you post the sentence in question, I can try to help you. Without it I would have to go hunting through the entire writing exercise myself. And that's something I'm not likely to do on a Monday morning.