PILOT (the Pathfinder for an International Large Optical Telescope) is a proposed 2.5-m optical/infrared telescope to be located at Dome C on the Antarctic plateau. The atmospheric conditions at Dome C deliver a high sensitivity, high photometric precision, wide-field, high spatial resolution, and high-cadence imaging capability to the PILOT telescope. These capabilities enable a unique scientific potential for PILOT, which is addressed in this series of papers. The current paper presents a series of projects dealing with the distant (redshift >1) Universe, that have been identified as key science drivers for the PILOT facility. The potential for PILOT to detect the first populations of stars to form in the early Universe, via infrared projects searching for pair-instability supernovae and gamma-ray burst afterglows, is investigated. Two projects are proposed to examine the assembly and evolution of structure in the Universe: an infrared survey searching for the first evolved galaxies at high redshift, and an optical survey aimed at characterising moderate-redshift galaxy clusters. Finally, a large-area weak-lensing survey and a program to obtain supernova infrared light-curves are proposed to examine the nature and evolution of dark energy and dark matter.