How did Greek mythology attempt to answer major life questions within ancient Greek society?
Any sort of faith beliefs arise out of a culture trying to make sense of a world that is not entirely known. For the Ancient Greeks, their mythology is full of examples that seek to identify deities with aspects of human life and the environment, so much so that it is impossible to detail every perceived example or characteristic that helped the Ancient Greeks explain their world in this answer. I will note some trends that are clearly present in Greek mythology.
The idea of fate is represented strongly in Ancient Greek mythology. Even the ruling gods themselves, the Olympians, were subject to fate and could not escape it. When one reads Homer, it's clear that unfolding events in the Trojan War were influenced by the gods and the heroes who fought, but the fate of Troy—destruction—was long set in stone. No matter the conduct of those involved, fate predetermined the Trojan defeat and Achilles' demise.
Another theme present throughout Ancient Greek mythology is how new generations of humanity usurped previous generations. This gives a sense of change over time and the underlying conflict between generations who forge their own path in life. Cronus, leader of the generation of deities known as Titans, who proceeded the Olympians, knew of a prophecy where he would be usurped in his position by one of his offspring (fate makes another appearance here). His son Zeus would then make this true with that associated myth. Of course there is also the other example of Oedipus, who tragically killed his father and married his mother.
Ancient Greek mythology was also characterized by patronage by the gods. Each deity patronized aspects of Greek life. Hera, wife to Zeus, was naturally a patron of marriage. Apollo patronized music, prophecy, and healing. Athena patronized organized warfare and therefore aided favored warriors. She even was the patron of the Greek city-state of Athens, which was named for her. This system of patronage gave meaning to the important aspects of life and the existence of these things is explained by the gods' patronage. The list goes on and on.
In summary, Ancient Greek mythology was clearly used as a means of explaining and even justifying aspects of Ancient Greek life. There are indeed many more examples that can be found upon further study. When studying Greek mythology, a good place to start is Ancient Greek literature, with the most obvious choice being the works of Homer. By reading works like those of Homer, one can truly understand how the Ancient Greeks viewed their lives through their mythology. Only then can we even begin to contemplate just how Ancient Greek mythology answered major life questions in Ancient Greek society.
Greek mythology asserts that individuals who struggle with major life questions have to be subservient to the divine in order to find beneficial answers.
Throughout Greek mythology, individuals found their answers to major life questions through subservience towards the gods. When individuals respected this structure, life's questions were answered for the better. For example, Baucis and Philemon struggle with the question of how they can remain together while the world is torn asunder. Their answer is in the eternity of nature, something that only happens because they showed respect to Zeus and Hermes. Odysseus's major issue is whether he will be able to go home. When he shows complete respect to Athena, his question is answered. In these examples, individuals who affirm the hierarchy of existence with immortals placed above mortals find their questions answered satisfactorily. Greek social structure was enhanced through its mythology.
When human beings challenge the natural order of things, the answers to their questions are not very pleasant. For example, Arachne has her major life question of how good she can be answered when she challenges Athena and an infuriated goddess turns her into a spider. Orpheus struggles with a painful question of existence in terms of how he can reclaim his beloved from death. He does not follow Hades's instructions and loses her forever. In both examples, individuals disrupt the hierarchical establishment in failing to adhere to the word of the gods.
Greek mythology clearly suggests that mortals who wrestle with life's pressing issues find their answers in the powers of the divine. Individuals must acclimate their own sense of identity toward the wishes of the immortals. When they offend Greek gods and goddesses, the answers to their questions become painful to endure. In Greek society, its mythology reinforces a terracing of power with the divine placed higher than human beings.