Is there something fundamental about the state of an atom or molecule
that, independently of other atoms or molecules of the same type,
determines if it is in a solid, liquid or gas state (plasma I
Not really. The idea of "state of matter" applies to collections of atoms or molecules, so a single atom or molecule doesn't really have a state of matter. When you do look at a group, then there is a definition you can use, it's the energy of that atom/molecule compared to the ones around it.
In the case of a solid for instance, there's no difference between the atom in that solid and the same atom in deep space. It's only when you get two atoms and compare them that you can say anything. In that case, you can compare their energy to the binding energy between them and that will tell you if they can bind into a solid, liquid, or separate entirely as a gas.
For instance, water has some self-attraction, you can see this as the surface tension. When you boil water all you're doing is adding random motion to the molecules. Eventually the average molecule has enough energy that it overcomes that attraction. With nothing left to hold it to other molecules it pops off the top and you call it a gas. But that atom didn't change.
Plasma is another issue, because that is defined as a state of atoms, and that's why I've never been comfortable calling it a state of matter. Plasma is a gas, pure and simple. The difference is that when you continue heating a gas, eventually the energy is not only high enough to overcome the inter-molecule binding, but even pop the electrons off them as well.
For plasma one might be inclined to say this is really a state of a single atom - does it have all its electrons. But you have to notice that that electron only gets popped off if it has an interaction with some other object with sufficient energy. A single atom in space has no interactions, so it would never ionize in the first place. And if you start with an ionized atom and its electron, they will quickly reconnect (EM is STRONG!) unless you keep interacting with it. So again, a single atom without its electron is an ion, but it's not a plasma, so we're back to collective behaviour again.
When a solid is dissolved in a liquid, is it still a solid
I want to touch on this too, because there's a useful subtlety here.
When you put salt in hot water, it dissolves. In this case the energy from the water is enough to overcome the attraction between the atoms, and it is pulled off the crystal to mix with the water. So you don't have water and salt any more, you have a mixture of Na ions, Cl ions, and water molecules. Or saltwater.
Now compare this to what happens when you salt and sugar. In this case the two happily sit together as a mixture, but they are not changed in form. They're also quite tasty, as salted caramel attests. If one of the two you mix is a liquid, like powder in water, then you have a suspension. In these cases the original materials remain in their original forms, just mixed together. This is different than the saltwater case, which is not simply salt and water in the same bottle.