Solid angle and surface area values in (photometric) raytracing
You have a lot of questions so let me take a shortcut and just try to explain how this is properly done (:

What you are interested in calculating is luminance (cd/m^2) for a pixel. A simple way you can do this is by using brute-force Monte Carlo integration over the hemisphere defined by the normal of a pixel, i.e. cast few thousand randomly distributed rays over the hemisphere. For each of those sample rays you evaluate the BRDF (1/sr) and multiply it with the luminance of the area light source the ray hits, or 0 if it doesn't hit anything. Sum up all these results and multiply by 2*pi/num_samples in the end, i.e. by the solid angle of each sample.

This operation gives you the the luminance for the pixel, which you then multiply by camera exposure factor to map to a proper range which you can tone map, i.e. calculate EV100 from camera aperture, shutter time and ISO-settings as follows:

float ev100=log2((aperture*aperture/shutter_time)*100.0f/iso);


And then calculate exposure factor:

float exposure_factor=1.2f*pow(2.0f, ev100);


For IES light profiles you can normalize the profile and modulate light luminance with the IES factor for the given direction the ray hits the light.