Well, interacting waves produce interference patterns. That shouldn't seem so strange. And it's good to keep in mind that wave-mechanical QM is based largely on classical wave mechanics.
I don't know exactly how Shroedinger came up with his wave equation, but maybe somebody here does.
You can treat the wave function as a mathematical probability wave because that's all that can be known for sure that it is. However, the fact that it actually works as well as it does seems to suggest that there's some more or less familiar wave mechanics happening in the underlying reality. But that might be misleading. I don't know. Anyway, probability distributions are waves, and the wave functions of QM are probability distributions.
Probability density function
I think you'll eventually understand it. And then you can explain it to me.
I don't think I've read the Fuchs article that I think you're referring to. Maybe I'll get to it this afternoon.