How did Earth and other planets form? Were planets formed in situ? Or are orbital changes relatively frequent? What determined the different deep layering of the solar planets?
Was there ever a collision of the Earth with another planet Theia, giving birth to our satellite? There is compelling evidence, such as measures of a shorter duration of the Earth's rotation and lunar month in the past, pointing to a Moon much closer to Earth during the early stages of the Solar System.
What is the long-term heat balance of Earth? How did its internal temperature decay since it formed by accretion of chondrites? How abundant are radiogenic elements in the interior? Did a "faint young Sun" ever warm a "snowball Earth"? 
What made plate tectonics a dominant process only on Earth? How did the planet cool down before plate tectonics? Was the Earth's crust formed during the early stages of its evolution or is it the result of a gradual distillation of the mantle that continues today along with crustal recycling? Is the crust still growing or does its recycling compensate for crust formation at mid-ocean ridges and other volcanic areas?
Can the now widely available topographic data be used to derive past tectonic and climatic conditions (in the multi-million year scale)? Do we know enough about the erosion and transport processes? Does the stocasticity of meteorological and tectonic events reflect in the landscape? How much has life contributed to shape the Earth's surface?
Can classical geomorphological concepts such as peneplanation or retrogressive erosion be quantitatively understood? Old mountain ranges such as the Appalachian or the Urals seem to retain relief for >108 years, while subglacial fluvial valleys under Antarctica are preserved under moving ice of kilometric thickness since the Neogene. What controls the time-scale of topographic decay?
What are the erosion and transport laws governing the evolution of the Earth’s Surface?[Willenbring et al., Geology, 2013] Rivers transport sediment particles that are at the same time the tools for erosion but also the shield protecting the bedrock. How important is this double role of sediment for the evolution of landscapes?
How resilient is the ocean to chemical perturbations?
What caused the huge salt deposition in the Mediterranean known as the Messinian salinity crisis? Was the Mediterranean truly desiccated? What were the effects on climate and biology, and what can we learn from extreme salt giants like this? How were the normal marine conditions reestablished?
There are several places where water does drain or has drained underground, but the destination is unknown. One example is "the Devil's Kettle" in Minnesota, a pothole which swallows half of the Brule River.
^Egholm, David L.; Knudsen, Mads F.; Sandiford, Mike (26 June 2013). "Lifespan of mountain ranges scaled by feedbacks between landsliding and erosion by rivers". Nature498 (7455): 475–478. doi:10.1038/nature12218.|access-date= requires |url= (help)
^Krijgsman, W.; Hilgen, F. J.; Raffi, I.; Sierro, F. J.; Wilson, D. S. (12 August 1999). "Chronology, causes and progression of the Messinian salinity crisis". Nature400 (6745): 652–655. doi:10.1038/23231.|access-date= requires |url= (help)