In The Epic of Gilgamesh, what questions/criticism does the writer raise? What is most important or effective about the reading, and do similar problems persist today? How or why, and what examples can you come up with?
There are certain themes and concerns that recur across the entire Epic of Gilgamesh. I would suggest that perhaps foremost among these themes is the intertwining of excellence and arrogance (and the self-destructiveness of that arrogance) that tends to be embedded within Gilgamesh's own characterization and partially divine nature.
Gilgamesh is a king; he is one-third god; he has super-human strength and physical capabilities. However, his pride is a weakness that frequently leads to disaster: he and Enkidu anger Enlil with the killing of Humbaba, and, later, Gilgamesh shames and castigates the goddess Ishtar to her face. After Enkidu's death, Gilgamesh's attempted quest for immortality represents a further variation on this same combination of themes: the very fact that he can even seek out the secret of immortality in the first place indicates his superhuman status, and yet his attempted quest must end in complete and utter failure, given Gilgamesh's own status as a mortal.
As to the second part of your question, that would require an element of thematic translation on your part, reimagining these concerns and ideas within a more modern context. For example, while there is no perfect modern equivalent to Gilgamesh himself (the partially divine warrior king of Ancient Uruk), there remains a great deal of inequality in modern society today. You might think in terms of wealth gaps and reimagine the modern-day Gilgamesh as a member of the wealthy elite or perhaps someone of celebrity status.
Another option would be to focus on politics and the themes of power. At the same time, you would also want to remember Gilgamesh's very real fallibility and human flaws, a consideration which, in a modern context, might bring up various scandals and the kind of behavior and mistakes that can ruin reputations and careers.