n is how much energy the electron has (the higher, the further from the nucleus)

I disagree with this description. It is true that higher values of n correspond to wave functions with higher energy, but higher values of l also correspond to wave functions with higher energy. n indicates the number of radial nodes.

A volume integral would be unitless

The volume integral has units of "number of electrons". Calling this unitless is unnecessarily misleading I feel, even if it is technically correct, since in physics we tend not to give units to quantities like this.