For Jupiter, if I recall, astronomers believe that it has a solid iron/nickel core similar to Earth's because of its magnetic field. As to the size of it, we have no clue, but it could very well be bigger than Earth since the Great Red Spot is three times the size of Earth.
As to the solid surface, the hydrogen would get compressed enough to form a liquid making an ocean surrounding the core. I remember hearing or reading something from a long time ago that the extreme pressure might even have the ability to compress the hydrogen into an almost solid state.
As for the heat that Jupiter produces, it is left over from the creation of the planets. Earth is the same way and that is why we have volcanoes. A good example of a "cooled" planet is Mars. At only half the size of Earth, it cooled quicker than the other planets.
For the other three gas giants, I would say they have the same characteristics as Jupiter, only with less pressure on their cores because of their smaller size. Uranus and Neptune might actually have oceans of liquid methane due their higher methane content (of which also gives them their blue color).
Interesting fact about Neptune is that it is actually producing more heat than when compared to Jupiter. Astronomers are actually surprised by this and believe that the winds on Neptune (strongest in the solar system) are a cause of Neptune's heat.
As for the "vast diamond" quote. In a planet, it won't happen (but it might actually rain diamonds on Neptune. Why, I don't know.), but in a star it can. There is actually a Guinness World record for the (possibly) largest diamond known to man that makes the core of a star. This would happen by the star creating vast amounts of carbon at the end of its life and having it become trapped inside the core. With hundreds of thousands of degrees in temperature and hundreds of thousands of pounds of pressure, we could see the formation of one massive diamond
Jupiter is thought to consist of a dense core with a mixture of elements, a surrounding layer of liquid metallic hydrogen with some helium, and an outer layer predominantly of molecular hydrogen. Beyond this basic outline, there is still considerable uncertainty. The core is often described as rocky, but its detailed composition is unknown, as are the properties of materials at the temperatures and pressures of those depths (see below). In 1997, the existence of the core was suggested by gravitational measurements, indicating a mass of from 12 to 45 times the Earth's mass or roughly 3%–15% of the total mass of Jupiter. The presence of a core during at least part of Jupiter's history is suggested by models of planetary formation involving initial formation of a rocky or icy core that is massive enough to collect its bulk of hydrogen and helium from the protosolar nebula. Assuming it did exist, it may have shrunk as convection currents of hot liquid metallic hydrogen mixed with the molten core and carried its contents to higher levels in the planetary interior. A core may now be entirely absent, as gravitational measurements are not yet precise enough to rule that possibility out entirely.
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