Xjyuk

1- Au cas où le projet ne serait pas fini cet après-midi, appelez-moi.

2- Au cas où il aurait échoué à l’examen, on lui donnera une autre chance.

3- Au cas où tu n’aurais pas retrouvé tes clés, Christian s’occupera de toi.

4- Au cas où ce cadeau ne vous plairait pas, dites-le-moi.

As far as I understand, the past conditionel in French is used to describe what could have happened if sth else happened in the past, and the events in sentence 1, 2 and 3 didn't happen in the past. Actually they haven't happened yet in the present so how can it be used here? Plus shouldn't the conditionel, in general, be used to describe a resultant action or event but all the events in those sentences don't seem like so to me.

From book Practice-Makes-Perfect : https://archive.org/details/practice-makes-perfect-complete-french-grammar/page/n33

First, there is a question of deciding what tense is used in the first sentence, or rather what verb, which will dictate what is really the tense; is it the verb "être fini" or the verb "finir"? In the first case we are simply dealing with the adjective "fini". The "present conditional" and "past conditinal" for these two verbs are as shown below.

                   present conditional         past conditinal

être fini       serait fini                         aurait été fini

finir             finirait                              aurait fini

The auxiliary for "finir" is "avoir", not "être" and it follows from that that the verb is not "finir" but "être (fini)"and therefore the tense is the present conditional" in the first sentence.

Your question is then valid for only "2" and "3".

2- Au cas où il aurait échoué à l’examen, on lui donnera une autre chance.

3- Au cas où tu n’aurais pas retrouvé tes clés , Christian s’occupera de toi.

You can use both the past and the present, but with a difference that depends, as always, on the context. If you use the present (au cas où il échouerait, au cas où il ne retrouverait pas), you imply that the action has not been completed in the present, that it is still going on or that it just hasn't been accomplished at all. For instance, the exam has not been taken yet, the person looking for the keys is still looking for them in a few more places, hasn't given up hope, etc. On the contrary when you use the past, what you imply is that the action has been finished in the past but that you don't know the result. for instance, in the case of an exam, again, the exam has been taken yesterday (for example) and you know that but you don't know the result; in the case of the lost keys it's a little difficult to explain that the action has taken place (that of not having found the keys, which is tantamount to having given up the search); it is considered as having taken place because a reasonable amount of time went by and that you can feel assured that the persons have done all that was possible.

“Au cas où” introduces a clause which is hypothetical in nature, and uncertain. Therefore, the conditional tense is used: “would _____” where the blank is filled with the verb in question. This is to further emphasize that “if/in case _____, then _____,” speaking hypothetically about a circumstance which could potentially happen, but not necessarily.

I’m not the best at describing it, but I hope this makes sense. In English, we don’t use the conditional tense in this way; it’s often present, followed by the future. Ah, the beautiful nuances in different languages. :)