Crude pomace olive oil is used because it is cheaper than using any other olive oil, and it doesn't matter what olive oil oil you use whether it is pure or crude. Crude pomace also gives the soap a green color which some manufacturers want.
Well these soapmakers use the boiled method in which they salt out their soap multiple times, thus getting rid of all the impurities. Castile soap was made with bad olive oil which was impure and cheap so that is somewhat similar to crude pomace. As for why the I think crude pomace olive oil...
This one guy that makes Aleppo soap in Turkey said they usually add Laurel at around the same time as the olive oil, but it takes practice to know exactly when.
Yeah I wonder if it will give any other properties to the soap made or not. Some sources also mention that Marseille soapmakers use only lye, a mix of sodium carbonate and lye, or just sodium carbonate these days so I'm not 100% on which is the correct way.
Two downsides of sodium carbonate is that requires 6-8 hours of cooking with normal hp, but since all boiling recipes require a lot of time it doesn't matter in this case. But since it is weaker than lye you have to add a lot more sodium carbonate, and Im not sure exactly how you can figure out...
Well I know Aleppo soap uses saltwater, but I'm not sure about Nablus. I think the longer cook times are due to how big the batches are and for Marseille soap they use sodium carbonate, which is significantly weaker than lye.
TOMH is saying all these soaps we are talking about are made using the fully boiled process, which includes boiling the soap for several hours or days, and occasionally adding salt or salt water to remove impurities and glycerin from the soap from the soap.