...L^2 and H have simultaneous eigenfunctions, so the angular part of the wavefunction is the eigenfunction of L^2 which is the SHs Ylm. l is the quantum number corresponding to the total angular momentum, m is the quantum number corresponding to its projection in a specific direction (usually z).

...So the solution, the wavefunction (from which you get the probability density) will be a product of a radial part and an angular part. Because L^2 commutes with H (it commutes with the kinetic term and the Coulomb potential since it's rotationally invariant being a radial function)...

Being a spherical system, the kinetic energy term includes the angular momentum operator (square magnitude of it, L^2), which can be seen when writing the Laplacian of the kinetic term in terms of spherical coordinates. Plugging into the S.E. yields a separable PDE with a radial and angular term...

Not a physicist, so my understanding here may be pretty flawed. Ignoring spin, the Hamiltonian of the hydrogen atom (energy operator, describes the dynamics of the system) is made up of kinetic energy and the Coulomb potential (electrostatic force between the electron and proton)...