I've read that higher energy means higher mass, and in atomic systems, the kinetic energy and potential energy actually contributes more mass than the actual particles themselves (or so I've read). So, how much of Earth's mass is created by the energy in the molten core? What would be the difference in mass between an almost identical Earth with no molten core and the Earth that we actually have?
Energy – How Much of Earth’s Mass is Created by the Energy of the Core?
earthenergygeophysicsmassmass-energy
Related Solutions
Circulating neutral particles will not by themselves create a magnetic field. However, if the neutral particles are moving through an existing magnetic field, and the neutral medium is conducting, then the magnetic field will induce a current via the Lorentz force. That induced current will in turn create it's own magnetic field, which may enhance the existing magnetic field. If things work out right you have a self-reinforcing dynamo where motion thru the magnetic fields drives currents and those currents in turn support the magnetic field. However, there had to be some sort of "seed" field to get the thing started in the beginning.
The earth's oceans do in fact generate a measurable magnetic field$^{[1][2]}$. As you have already pointed out, the motion of charged particles generate magnetic fields, so it makes sense that the earth’s oceans would do the same. In fact, the oceans make a contribution (albeit a small one) to the Earth's overall magnetic field.
The moving salts within the oceans have electrical charge which means you have electrical currents, and since the oceans move in cycles, the motion of the tides etc., as you pointed out, the oceans contribute to the total magnetic field of the earth.
In the image below, we see how this magnetic field is distributed about the northern hemisphere with the United States and Canada in the center of the sphere, and how its strength varies at different points. The European Space Agency in 2013 launched three satellites, a system called Swarm, which was designed to study the earth's magnetic field in detail and was also used to map the magnetic field emanating from the oceans.
As can be seen, the ocean generated magnetic field is on average $(1 \ \text{to} \ 2)\times 10^{-9}$ Tesla at sea level. This field goes to roughly $10^{-9}$ Tesla at the height of about a few hundred kilometers, or average satellite height. This means that this magnetic field is about $20,000 \times$ smaller than the Earth's magnetic field ($\approx 40\mu$Tesla) caused by the motion of charged particles in the Earth’s core.
References:
Analysis of Ocean Tide-Induced Magnetic Fields AGU Journals, 08 November 2019.
Ocean Tides and Magnetic Fields A short video by NASA and links therein with other interesting magnetic effects of earth's oceans.
Best Answer
According to Table 2.17 from page 109 of Chemistry of the Climate System by Detlev Möller, the heat content of the inner core of the Earth is $\sim 3.6\times 10^{30}$ J, and the outer core is $\sim 1.5\times 10^{31}$ J. The total heat content of the Earth is $\sim 2\times 10^{31}$ J. The author stresses that these are only crude estimates based on theories that give mean temperature and composition for the various layers.
Using $E=mc^2$, the mass equivalence of the inner core heat is $\sim 4\times 10^{13}$ kg, the outer core is $\sim 1.67\times 10^{14}$ kg, so the total for the core is around $2.1\times 10^{14}$ kg.
For comparison, the Earth's mass is $\sim 5.9722\times 10^{24}$ kg. So the core heat contributes around 1 part per 29 billion of the total mass.
Here's the contents of Möller's table.
It's surprisingly difficult to find this geothermal energy data. Wikipedia gives a figure of $10^{31}$ J for the internal heat content of the Earth, linking to a report which quotes a figure of $12.6×10^{24}$ MJ from What is Geothermal Energy by Dickson & Fanelli (2004), but that article gives no details for the calculation.