Short answers, pulled from the story.
Planetary differentiation is the process where high-density materials sink through lighter materials while low-density materials rise to form distinct layers. This tendency occurs when temperatures are high enough for both materials to become plastic or molten, allowing iron and siderophile elements to congregate toward planetary interiors.
Heating from radioactivity, impacts, and gravitational pressure melted parts of protoplanets as they grew into planets. The short-lived radioactive isotope 26Al was probably the main source of heat driving these early changes in asteroids like Vesta.
Earth has an average density of 5515 kg/m3 because it retains a large iron core, whereas the Moon formed from material splashed into orbit by a collision that removed most silicate material but left dense metal behind. The Moon's density is substantially less than Earth due to its lack of a large iron core.
Core formation utilizes several mechanisms including percolation, diking, diapirism, and direct delivery of impacts. Percolation involves the movement of metal downward driven by the metal-to-silicate density difference, while diking forms new rock within fractures when sufficient pressure overcomes fracture toughness.
Study of trace elements in igneous rocks reveals which minerals were lost from the melt and how much source rock melted to produce magma. Research on Hawaiian lava lakes showed that crystals formed within magma fronts demonstrated differentiation through chemical melt of phenocrysts.