I'm new to quantum physics, too. But here is my take:
One. For a nonrelativistical particle, assuming only one, the wave equation can be solved using Schroedinger Equation, given a certain potential energy distribution. Many times, there are many eigensolutions to Schroedinger Equation, given a potential. In this case, these eigensolutions form a vector space of solutions (Schroedinger Equation is a linear differential equation after all), and any linear combinations of these solutions can be a solution. With certain boundary conditions, one can often determine a unique solution. Each eigenstate composing the entire solution has a certain probability showing up when a measurement is made, where the probability is the magnitude square of its linear coefficient, given that all eigenfunctions are normalized.
Two. I'm not sure whether what you said was completely right. After all, definitions of a "particle" or a "wave" are ill-defined. But if you want to attribute "existing everywhere" to a wave and "being local" to a particle, then as I mentioned above, before a measurement, a particle doesn't have a definite position or energy or any macroscopic physics parameter. That's wave-like. After a measurement, you've collapsed the wave function, thus only one eigenstate will describe the behavior of the particle. This is particle-like. As for why do measurements have such "magical" powers to a particle, this is still a open question. Classical Copenhagen Interpretation explains measurement problem this way: when a measurement happens macroscopic observation devices will have to infuse new energy to the system, thereby localizing a particle's physical property. But does infusing new energy cause the collapse of wave function? I am confused, too. Maybe I will learn it next year.
--Only a college freshman speaking. Hopefully this helps.
