What do you use a house for?
The biggest question is: what do you need a house for? what are the social, environmental or psychological reasons for having a house? Humans have houses for many reasons, and your merfolk may have different ones. You may come to find merfolk need NO houses, or live in dorms, or use individual pods. Form follows function, so ask why they have houses.
- Humans have houses to protect from the environment, but anyone dealing with depths is likely to be more temperature-tolerant. But merfolk may need temperature-controlled environments for special functions like reproduction or child-rearing. Places may need to be sheltered from currents, or create still areas where debris can settle out of the water for whatever reason. The nature of water makes temperature control tricky if water can circulate. I don't know what heat sources merfolk might have, but keeping spaces warm might involve considerable complex engineering.
- People want houses to protect from violence. That might mean saltwater piranhas, giant octopi, or rival warring merfolk. the shape of the dwelling will be defined by what it is you are trying to keep OUT. Fine woven mesh protects from small things, while huge stone spikes might be needed to keep out a Kraken.
- Humans use housing for privacy. They live close to other people, but don't want others to watch them all the time. For visual privacy, simple woven mats might suffice. Sound travels oddly in water, so soundproofing might look very different for merfolk. The bigger the community, the more merfolk might not want strangers staring at them. If merfolk evolved as nomads and live now in cities, pretending your neighbors AREN'T next door might be crucial. Or, maybe they have no privacy in families but keep out strangers. Do your merfolk value secrecy? Do they commit crimes, defy the government, or live in communal utopia? Do the rich merfolk want to conceal their wealth so as not to anger others?
- Humans use houses to store stuff they don't want lost/eaten/stolen. Keeping resources contained is an undervalued thing. Do your merfolk steal? Do fishes sneak around trying to eat their food? What kind of threats do you need to deal with for your stuff?
- People want houses to show off. Perhaps rich merfolk build houses out of stone because it's difficult, not because of practicality. Keeping up with the Joneses is a big deal, and if you're from a rich family, you might be more reproductively successful. Or merfolk may desire to impress humans, despite not caring for houses themselves. They may want houses because they imitate humans whom the perceive to be rich, successful, or influential.
- Houses are places to interact in controlled environments. Do your merfolk mate publicly? privacy for mating could be important. Do they need someplace to contain sperm so it doesn't randomly impregnate strangers? Do they invite humans over? If so, dry or semidry parts of a home are crucial. A half-flooded top floor could be a hosting place. Do they have special work functions (like smelting on the surface, while sitting protected from heat in the water)? This will radically affect how these places are shaped.
Materials:
People build first with what they have, then with what serves the function. Chipping stone in an aquatic environment might be a high-resistance task, but saws would have continuous cooling. Metal tools could be expensive or tricky. Wood often rots more slowly in the water, but could be expensive and vulnerable to parasites. Growing corals and plants to fit needs would be an elegant solution to many problems, but your merfolk would need a fair amount of biological technology to do so. However, given an understanding of selective breeding, they could guide various types of mangroves, corals, kelps, and the like to do their bidding.
Shells might function as direct components of walls, OR the calcium carbonate could be used for some type of cement. Perhaps the glues used by some kinds of clams and barnacles could allow walls to be grown from these organisms, or use these organisms to bind together other hard materials into solid structures. A barnacle bred to glue things on multiple sides could be the binder for hard material in the environment, from shells to loose stones. Then the irregular parts get filled in with whatever is available.
Design:
Your merfolk might extend home up to the surface, and then have no roof. homes might even be hanging down into the water from floats, like curtains weighted with stones or shells. A home then becomes mobile or even migratory like a ship, as needed. Such houses would need an anchor when staying put. If moving against the current but with the wind, sails could be used to guide the house. You could even use them to alternate direction, 'sailing' your town back and forth to keep it in place or following harvests.
I'm guessing due to buoyancy, a lot of the structures would be made of seemingly flimsy materials like kelp. Flexibility might be more important than strength. Solid structures make more sense to keep out predators like sharks and sea monsters, but a lot of big things aren't going to push through barriers they can't see through.
