Why is it that on a natural reef, there is way more fish poo floating around than in the dirtiest reef tank that you ever saw, but yet still, the corals feel the need to capture food with a higher nutrient profile at night?
It seems as though you're assuming that the only function of polyp extension is food capture. Yes, this is a major reason a coral extends its polyps, but certainly it is not the only reason. Besides, there is a lot more food available at night and some corals extend their polyps 24/7. There are many reasons both day and night to extend or not extend polyps and food availability, while critically important, is only one of them.
What about the inorganic nutrients (ammonia) that the fish leave behind? Doesn't it just get converted to carbon/sugar/carbohydrates/mucus that is not all that useful for growth?
Ammonia can be used to produce amino acids, proteins, and other N-rich substances, not sugar/carbohydrates. Some N is lost as mucus, and dissolved substances, for sure, but symbiotic corals are pretty good at retaining nitrogen.
Or, can some corals culture (with mucus) bacteria to eat, and in the process convert undesirable nutrients into something more useful for growth?
Corals probably can and do make use of the bacteria on their surface layers (actually, it would be really shocking if they weren't living in some sort of symbiosis) but it doesn't seem that these surface layer bacteria can contribute very dramatically to the nutrition of the corals. Also, which nutrients are you suggesting would be undesirable? The zooxanthellae can take up ammonia, nitrate and orthophosphate and translocate these to the coral. However, the concentrations in nature are usually so low that this only happens (net uptake) when the local conditions elevate these inorganic nutrients for short periods of time (upwelling, passing fish, etc.).
There may be one other step in the process: the bacteria in the sand, and the bristleworms. I have heard it said that without the worms, there would be far fewer corals, since the worms break down poo into a finer particle, and may process out some of the inconvenient chemicals.
Large decomposers breaking detritus into smaller pieces definitely faciliates decomposition and may make detritus a much more managable size to eat for a lot of corals. What 'incovenient chemicals' are you talking about though?
cj