Are there any other measurements that have contributed to current understanding of Earth's core besides these two?
The answer is of course "yes". Other answers have already alluded to laboratory experiments that attempt to re-create conditions similar to those well inside the Earth.
I'll provide two others; there are many more.
One is radio astronomy. Determining the apparent locations of quasars has drastically increased the accuracy of the Earth's orientation. Doing this in conjunction with modern communication techniques results in Very Long Baseline Interferometry. The combination of the two has reduced the uncertainties in the Earth's orientation to well under a milliarcsecond. This gives deep insight (pun intended) into the nature of the Earth's core. The Earth's Chandler wobble does not behave quite like that of a rigid body. How this varies over time gives insights into the nature of the Earth's core. The Earth's free core nutation is also observable from the precise Earth orientation parameters.
Another is precise gravity models of the Earth. These too give insights into the Earth's core, including the Earth's moment of inertia, the Chandler wobble, and the free core nutation. Going beyond the Earth, gravity models provide one of the key observational techniques for studying the interior of the Moon, Mars, and Jupiter. Scientists know that the Moon and Mars have partially molten cores thanks to gravity models developed from precise orbit determination based on the many satellites that have orbited the Moon and Mars. Scientists know that Jupiter has a diffuse core thanks to precise orbit determination of the Juno spacecraft's orbit about the planet.