The consensus view is that that the Earths core is composed mainly of iron alloyed with about 10 percent nickel and traces of other heavy metals, but I have seen at least one account which says it is partly iron sulphide. Has the iron sulphide theory been discredited, or is it a view still held by some geologists? There is a recent theory that about 5 percent of the core is composed of silicon. How valid is this, and will we ever be able to establish more precisely what the core is made of?
We know the density of the core from seismology.
We also know the density of pure iron at given pressure (P) and temperature (T).
From these two, people have long noticed that there is a discrepancy: the actual core is lighter than a theoretical core made of pure iron. This is known as the Core Density Deficit (CDD). To explain this CDD, we need to add light elements in the core composition. There are many candidates, such as sulphur, carbon, silicon, oxygen, etc.
This is actually needed by geochemists, who try to relate the Earth's composition to that of chondrite. They find that the Bulk Silicate Earth (BSE = crust + mantle) is depleted in some elements compared to the chondritic reference, and have to invoke a hidden reservoir to explain this. For some elements, this reservoir could be the core.
To further constrain this, petrologists run some high-P/high-T experiments to see the behaviour of elements during the Earth differentiation. Some elements are said to be lithophile, meaning they will stay in the BSE, while other elements are considered siderophile, hence should have joined the metal phase during segregation of the core. The problem is that some experiments disagree with others, as the behaviour of a given element can change depending on the conditions. For instance, niobium and tantalum are considered lithophile, but can be siderophile in reduced conditions.
I'm afraid the bottom-line of all this is: there is no simple answer to this question...
No, simply because we cannot access it and the methods for understanding it are essentially indirect geochemical and geophysical measurements (mostly experimental petrology and wave propagation experiments). We only talk about models, so far. So much so anything deeper than the crust - to be more precise, anything but upper continental crust is either measured indirectly - i.e. we have no access. The mantle? Mostly theoretical with the help of xenoliths (rocks from the mantle that are brought up by ascending magma).
Simulating the conditions in earth's core is difficult. Maybe one day it can be done. Until then, we must live with indirect methods.
The sulphide fractionation towards the core is a valid hypothesis. https://www.geochemicalperspectivesletters.org/article1506
The "lead paradoxa" express another inequality in earth's fractionation, if compared to for example meterorites.