Did I understand the double slit experiment correctly?

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Do I get this correctly: if a wave/particle is not measured, it is in the wave state, and if it is measured, it becomes a particle, fixated on an eigenstate of the probability distribution of the wave function.
So the universe - matter, light, each natural phenomenon - if unmeasured, is in a wave state, and if measured, in a particle state?
 
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I wouldn't think of it as "particle state" vs "wave state". The particle isn't actually in a single state until it is measured/observed, so there really isn't a "wave state". The idea is that because the wavefunction of the particle describes all possible states the particle may be in and gives the probability of finding the particle in any of those states. Note that a "state" means that if we measure the particle, we will find that it has a particular value for momentum, position, velocity, etc. A different state would have different values.

Also, I'd just like to mention that a measurement isn't the same thing in quantum physics that it is in everyday language. If a particle interacts with something in a irreversible way, then that causes the particle's wavefunction to collapse and the particle enters a particular state. It doesn't require that some measurement apparatus like a microscope or detector be present.
 
Eh maybe! You should think of it this way. There is probability amplitude wave associated to every particle. So, everything is a particle and the reason they behave like waves is not because they are wave-like (as far as quantum mechanics goes today), but because there is probability amplitude associated to its position and momentum. We call this wave, wavefunction. For example, in quantum mechanics, light is not a wave, it is a collection of particle called photons with associated wavefunctions, which is related to electromagnetic waves. I hope I didn't confuse you.