In FE, there'd be no (or few) problems with Surface, Love and Raleigh waves. It's the Primary and Secondary waves that give it trouble.
The density of the cores are the only major assumptions, but even then they have been calculated and peer reviewed, so are probably fairly close. Once scientists figured out (don't ask me how...) the size/weight of the earth, they put in the densities of the crust and mantle (both of known density) and what was left must be the core. Iron and Nickel seem to fit the data, and would also give the magnetic field around earth.
Seismics are often used to find densities of things, and map out locations when they are underground. It's a well documented and reliable technique.