What are some of the “smaller” questions?
(these are but a few arbitrary choices)
How does the Sun heat its corona? 
How do rotation and mass-loss affect the evolution of stars, and how big does a star have to be in order to eventually explode as a supernova?
Why isn’t the supermassive black hole at the center of the Milky Way giving off lots of energy, like we see in quasars?
What are gamma ray bursts?
What is the history of star formation in the universe? When were the first stars formed?