Excellent answers from the physical sciences. Let me add a few known unknowns from the humanities. These tend to be less clearly stated, but the following questions account for a big chunk of what I consider fundamental research in the humanities.

1. Do political systems inevitably converge to liberal democracy? Francis Fukuyama answered "yes" in The End of History and the Last Man. Samuel Huntington answered "no" in The Clash of Civilizations as did his student Fareed Zakaria in The Future of Freedom. I haven't yet made up my mind. This question may seem like it's all about opinions and debates, but there are actually fundamental models and first-principles analysis that can be applied here. People even build simulation models to explore this.
2. Is world peace possible, given the constraints of human biological nature? Darwinian models describe us as a species that represents a design trade-off point between cooperation and competition, with traits that help us do both (like empathy for the former, ability to do political manipulation for the latter). It is increasingly clear that to survive as we create an increasingly techno-political integrated society (THAT possibility depends on Q1), we'll need to move unnaturally far towards the cooperation end. Can humans actually stay there?
3. What are the true limits on human potential and have we reached them yet, or have we already peaked sometime in the past and are on a decline path? Given our DNA and design, and current understanding of limits in narrow areas like chess and lifting heavy weights, what are the true limits on human potential? What can we never know or do? (you can ask individual/collective versions of this question).