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There is much known about Earth's core from painstaking analysis of seismic data, and from detailed magnetic field maps and trends over time.
Are there any other measurements that have contributed to current understanding of Earth's core besides these two?
Short answer: yes, most of our knowledge is not from seismics and geomagnetism, but from experiment and measurements, deductions from geochemistry and -physics, material science and physics in general (besides the already mentioned measurements of earth's gravity, but the variations are rather connected to mantle convection, water storage, etc. than the core). But all in all those weigh much more than "painstaking" seismics and geomagnetism.
One experiment to replicate pressure/temperature (p/t) conditions is the diamond anvil cell and its cascaded version in combination with laser heating of the probe, to obtain data about mineral and cristalline phases and transitions under conditions in a planet's core.
Wikipedia: Diamond Anvil Cell
Teaching Mineralogy; Topical Resources The Diamond Anvil Cell (DAC)
And some random links on the outcome of the matter:
Reviews of Geophyics: High-Pressure Experiments and the Phase Diagram of Lower Mantle and Core Materials
Nature Communications: Implementation of micro-ball nanodiamond anvils for high-pressure studies above 6 Mbar
Earth and Planetary Science Letters: The Earth's core composition from high pressure density measurements of liquid iron alloys
Geophysical Research Letters: Iron-Nickel alloy in the Earth's core
Laser pulses is a developing technique to reach very high p/t conditions, higher than diamond anvil cells can, but they do not last that long
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.
I think the answer is 'no', in the sense that 'seismology (i.e., the study of waves propagating within the Earth) tells us the details in a far superior way'. See this paper for an informative treatise: Brush, S. G. (1980). Discovery of the Earth’s core. American Journal of Physics, 48, 705-724, and these lecture notes http://www.geo.uu.nl/~berg/geodynamics/lecturenotes.pdf on Geodynamics.
Interesting additional tools that you'll find in these works are:
Figuring out the mass of the Earth (using Newton's laws). People found that the average density of the Earth had to be $5500$ kg/m$^3$, while rocks at the surface only had a density of about $2700$ kg/m$^3$, so the interior of the Earth had to be of higher density. Under the assumption that pressure alone cannot be responsible for the increase in density, the high densities had to come from chemical changes such as the presence of metals, particularly because of the Earth's magnetic field. Under the assumption that the pressure was responsible for the high density, the Earth's interior could be gaseous, for example, plain air is more compressible than just rocks, so the Earth's interior could be air (as proposed by Benjamin Franklin in the US!).
A second clue that the Earth had to have a radially varying density came from the Earth's moment of inertia (which was a known quantity, based on astronomical measurements such as the Earth's axial precession), which makes it possible to create models that relate density and radius. Of course, with neither quantity well-known, you couldn't do too much with this. But assuming an iron core (with known density, and responsible for the Earth's magnetic field) overlaying a mantle (with density close to that of the Earth's surface), the first radial profiles could be made.
Models could be made that concerned themselves with the rigidity/deformation of the Earth under forces, such as the response to ocean tides, terrestrial tides and the Chandler wobble. These could be used to even construct some radial profiles of some approximate elastic properties. Only in the 1960s, after seismology was a firmly established technique, people additionally found additional non-seismic evidence of the Earth's eigenvibrations/free oscillations where the Earth (as a whole) rings like a bell, which was a very useful independent tool to verify layered profiles of the Earth's elastic properties, and which were important in the discovery of, e.g., the solid inner core.
The rigidity (from the previous point, and mentioned in the first point) can be linked to densities using (thermodynamical) equations of state. Hence, figuring out the appropriate relations between density/pressure/rigidity was of large importance.
Similarly, with the assumption that the Earth was made with the rest of the solar system, it was possible from the spectral lines of the sun and the chemical makeup of meteorites to figure out the likely bulk chemical composition of the Earth (https://en.wikipedia.org/wiki/Chondritic_uniform_reservoir). If you additionally know the pressure, density and rigidity information, you can make relatively informed guesses about candidate materials that must be present. That is, similar to the answer by a_donda, one must still test how materials really behave in these strange pressure and temperature regimes, which is mostly laboratory work using, e.g., a Diamond Anvil Cell.
Similarly, these lab experiments can tell us that for a given pressure on a given material, what the expected temperatures can be.
But if you want very solid answers about the 'nature' of the Earth's core, all the most direct answers come straight from seismology. That is the best method to actually constrain the location of the (elastic) properties within the Earth, at a much higher accuracy than any of the other methods can. As secondary consequences, we can use these measurements to make chemical hypotheses about the makeup of the core etc., but those are entirely based on the assumption that the seismic data is correct....so they're not an 'independent' kind of measurement/model!
The comments to the other question have already alluded to some physical answers that also come out of the lab: One might think that the heaviest elements in primordial Earth should have all sunk into the core by now. But that's not actually the case: For example, uranium with a density 2.5 times that of iron (19.1 vs 7.9 g/cm3) has an abundance of about 4 ppm in the mantle, whereas in the core its abundance is believed to be essentially zero. How do we know this? We can make experiments about how easily uranium dissolves in rocky materials vs in iron melts -- and it turns out that if it has a choice, uranium very much prefers to associate itself with rocks and not with iron.
