The irradiance falloff happens at this exact ratio of r^2 because irradiance is the power per unit area incident on a surface point. The 4 pi r^2 is looking at the area of the circle around.
Is it 4pi instead of 2pi because we're using solid angles instead of a regular angle?
I don't think this is related to solid angles. I believe it is 4π4\pi4π because the equation for the surface area of a sphere is 4πr24\pi r^24πr2. For the inner irradiance, r=1r=1r=1 and thus it is just 4π4\pi4π.
Why can we use intensity to describe irradiance here?
It can get a bit confusing when we mix colloquial lighting language with these terms, but in this case, "intensity" is just a way of expressing how bright the light will be at different distances.