Mostly iron because Iron packs really well under extreme pressure and temperature, which is really what drives the segregation of materials inside a planet. It is a matter of space/volume and not all compounds pack to the same extent so unit volumes vary, and will change even for a given compound as P and T vary. The amassing of iron in cores is just a response to the existence of pressure and temperature change with depth (other elements or compounds are less compatible with high pressure, basically, so they end up closer to surface than things that deal well with high P and T, and iron deals the best of them all, apparently.

On top of that density consideration are the chemical needs of each element. Most elements are not all that stable except when in a compound, but iron actually is fairly stable chemically, as a native element. Plus, there is a lots of iron around, with iron being the most stable element made by stars (heavier elements want to break apart, undergo fission, and lighter ones want to combine and make bigger atoms, undergo fusion).

The physical characteristics of iron at extreme pressure and temperature are not well known because it is really hard to study something several thousands of kilometers below a mass of silicate materials (study in place) and it is difficult to make even in laboratory, and then only in very tiny amounts, with even a problem of time duration coming into play (it won't stay the same if you ease off pressure or temp).

People are working on it. More near the frontier of knowledge than a well-characterized material. Even the nature (PT-conditions) of the transition from face-centered cubic to hexagonal close-packed structure is poorly defined from what I have seen about it.

Including minor nickel to make some sort of alloy, and its effects on behavior, is also still poorly understood.