Understanding the irradiance map generation
Yes, the convolution is effectively evaluating the following integral:

$$
\text{irradianceMap}(\vec n) = \frac{1}{\pi} \int \text{environmentMap}(\vec \omega_i) \, \max(\vec n \cdot \vec \omega_i, 0) \, d\omega_i
$$

This convolution isn't about whether the environment map is equirectangular, cubemap, or some other projection. We still have to do the convolution regardless, because we want to get the total diffuse light (irradiance) coming to a surface with a given normal, from all directions in the hemisphere around that normal. The environment map to start with just has light from one direction (radiance).