None of the solutions that we use on land will be applicable for merfolk in a described setting (classical antiquity technology, 100-200m below sea level). Waste management should be based on water filtration to prevent the spread of toxic products of waste decay. Most importantly, all solutions should be 3-dimensional because in aquatic conditions waste and products of decay may stay in the water column.
All waste that a merfolk settlement will most likely produce can be roughly divided into 3 categories:
- inorganic matter that does not dissolve in water (or does not dissolve quickly enough to make rubbish a non-issue);
- organic waste (all organic matter excluding excrements);
1. Inorganic matter
These include things like pottery, stone, and metal. Some of them will eventually degrade (some metals) due to saltwater corrosion, some can last for centuries (stone).
This type of waste can be buried in a marine equivalent of a landfill. However, it is a bit too wasteful. A better solution would be to break things into smaller pieces and use them for filtration (more about it below). Porous stone is one of the best natural filtration media as it has a large surface area where beneficial bacteria can live.
Metal is a bit harder to deal with. Perhaps, merfolk can collect scrap metal and sell it to humans. If this is not an option, just bury it somewhere. Please note that copper is highly toxic to some aquatic animals, for example, crustaceans (shrimp). If bronze corrosion can lead to increased concentrations of copper it should be considered in your urban and waste management planning. Most likely it is not a problem, but I would suggest verifying this point (I am not a chemist, please ask someone more qualified).
2. Organic waste
Organic waste becomes a problem only when merfolk settlements grow too big and natural filtration mechanisms fail to keep up.
Organic waste in water can become a toxic nightmare, especially when water flow and biological filtration are not sufficient. Cities must be planned to avoid 'dead spots' where water hardly moves and debris accumulates. All organic matter decays underwater and produces toxic compounds. Ammonia is one of the most dangerous for aquatic life.
Some of the approaches to managing organic waste would be:
- build homes with good water flow (this also might be a necessity if merfolk biology is sufficiently close to sharks and they have to sleep within currents to keep breathing);
- plant live corals on all walls (corals receive nutrients from the water column);
- keep bottom-dwelling fish and crustaceans as waste cleaners (can be done in a form of a farm or housepets);
- build filter chambers filled with edible algae and small aquatic animals (imagine a waste bin with walls made of coarse filter material [remember those stones and pottery?]; water should flow through the walls for filtration to work);
- plant fast-growing aquatic plants inside settlements and around them (most aquatic plants can receive nutrients from the water column; you need to research what plants can thrive at your desired depth).
For public buildings which produce more waste, it might be necessary to supplement the abovementioned options with off-site facilities. Something like a big coral garden (perhaps, inside a cave with complicated currents) with fish and small animals would work wonders. Alternatively, waste can be used to fertilise farms (think 3D, make sure you have mechanical filtration around farms).
It is very important to remove organic waste promptly. Organic matter decays very fast (in a matter of hours) and turns into a slimy substance that is hard to transport or clean. In aquaria, this decayed matter is usually removed with the help of siphons. But I do not think that it is a plausible solution in your conditions.
First of all, you need to decide whether merfolk share human sensitivities when it comes to urination and defecation. Humans perceive excretion as a dirty activity that should be done privately. This is not necessarily the case for other beings. It is also not clear if merfolk are capable of regulating the excretory system the way humans do. In other words, I am not sure if merfolk can hold long enough for a trip to the toilet.
If merfolk need privacy, the best solution would be fish gardens in small chambers or caves. Shark faeces are semi-liquid and contain some half-digested food. Fish will eat this food. The rest is a perfect fertiliser for corals.
I did not find a lot of information about shark urine (because different shark species manage urination slightly differently; and yes, most sharks urinate), but most marine fish produce highly concentrated urine that is highly beneficial for corals, algae, and other tiny aquatic organisms.
It might not be very acceptable from a human point of view, but the smartest solution for merfolk would be having 'vegetable gardens' instead of toilets. Excrement collection is not a very feasible option, and filtration would waste a lot of nutrients. So the best option is to fertilise your own garden.
This answer is already very long, but it still may be missing some details. Feel free to ask in the comments if you need me to elaborate on something or to supply additional information.
I would also encourage learning about aquarium filtration and natural aquatic ecosystems to have a better understanding of various processes happening in aquatic habitats.