Well as a geologist, I feel obligated to try to spread some light...
Naryar and such are basically correct... "Magma ocean" is a sort of fuzzy term, and it generally means whatever the speaker wants it to mean. The inside of the Earth is what DF might call 'semi-molten rock'. It's sort of squishy, like really tough silly-putty or carmel, but the pressure is too high for it to actually melt and flow quickly. The stuff that comes out of volcanoes on Earth is usually the most fluid bits of the mantle that have essentially been distilled off by heat. Io, on the other hand, apparently has a layer of the mantle that's hot enough and at low enough pressure to actually flow freely.
(For those who care, I believe the technical distinction between 'runny rock' and 'stiff magma' is the amount of actual crystals in the mix, and Earth's mantle is mostly crystalline. The crystals bend and break under pressure which let the material flow, but there's much more crystal than melt in the mix.)
But I digress. This is cool is not only because it's there, but because a layer of magma 50 km thick is what official science jargon calls 'a whole buttload' of magma. To the tune of about 8% of Io's volume. Just about the right amount of magma to fill all of Earth's oceans, in fact. Hmmm...
As Mercury and Mars are well and truly tectonically dead - cooling linked to size/surface area. No one is really sure about Venus, or about what mechanism has pevented Earth's innards cooling - possibly due to some property of the core.
Nah, Earth is just big and full of radioactive stuff generating heat, so it stays warm. Surface area to volume, as you said; Earth is bigger, so it cools off more slowly. Venus too; Venus's surface is covered in volcanoes which are reasonably new, and we have no reason to imagine more couldn't erupt. Mars is sort of an edge case; parts of Olympus Mons are supposedly 100 million years old or so, which is pretty young compared to the 4.5 billion year age of the planet. So it's possible that Mars still has some heat left in it that will, in maybe a few more million years, make enough melt to create a new volcano. And Mercury we don't know jack about,
yet.
In fact... if you want to consider Boatmurdered-esque deployments of magma, Venus is the place to go. As far as people can tell, about half a billion years ago the entire surface of the planet was flooded with retarded amounts of magma, which gradually cooled into basalt. People think that it had something to do with Venus not having plate tectonics, and so the heat being generated inside it built up and built up until the crust basically melted and sank into a... well, a magma ocean. Which is probably as good as the story is going to get until we land there and start digging, and excavate small steel helmets and pickaxes from the rock...