I have recently started studying quantum mechanics, and here are some things that are really confusing me.

1. Particle in a box: Supposedly, the square of the magnitude of the normalized wave function gives the probability that the particle will be in a specific location. I understand that for the superposition states, the states add up so that the wave function of some particle that can be found in some superposition of eigenstates with various energies, such that the eigenstates have wave functions $\psi_1, \psi_2, ...$, then a particle whose initial wave function is of the form $\sqrt{p_1}\psi_1+\sqrt{p_2}\psi_2+...$ will be found in the eigenstate corresponding to $\psi_1$ with probability $p_1$, the eigenstate corresponding to $\psi_2$ with probability $p_2$, and so on. But, if you took the square of this superposition wavefunction in order to get a probability distribution, it won't, in fact, be the same as the simple superposition of the original probability distributions with weights according to the probabilities, because there are cross terms. The explanation I've heard for this is that the integral of each cross term over the width of the box is zero, so that the wave function can still be normalized. However, the cross terms can very much be positive at some locations and negative at others, so that the resulting probability distribution from squaring the wavefunction is different from what you'd get from making a superposition of the wave functions of the eigenstates according to their probabilities.

2. Normalization of the hydrogen atom wave functions: I'm struggling to understand how the $r^2sin(\theta)$ term (which I conceptually understand represents the ratio of volume between a voxel in spherical coordinates and a voxel with the same dimensions in Cartesian coordinates. In particular, how does it affect normalization? For instance, if you already have a normalized radial wave function $R_{n,l}$ and a normalized spherical harmonic $Y_{l,m}$, it is said that the product $R_{n,l}\times Y_{l,m}$ will be a solution to the eigenstate wave equation $\psi_{n,l,m}$. That being said, since $R$ and $Y$ are both normalized, and there is no overlap in the variables upon which they depend, their product should also be normalized. Except, there is an $r^2sin(\theta)$ term in the integral for the product, since we are integrating over volume in spherical coordinates. The $sin^2$ was already accounted for in the normalization of the spherical harmonic, but, in light of that, would the normalized wave function simply be equal to $\frac{R_{n,l}\times Y_{l,m}}{R}$, in order to account for the fact that, if the particle has an equal probability of appearing at two different radii, and one is farther out than the other, the density w/ respect to volume will be lower at the farther radius (and to cancel out the $r^2$ term in the normalization factor)?