Here is my take on your questions, based on the literature search I have done. I am not personally experienced in "boiled" soap, so I'm just passing on what I've read.
1. Is only half the lye solution added because you want to reach "trace" faster? About how long time into boiling are you adding the other half of the lye solution?
Adding the lye in several batches to a hand-stirred "boiled" soap, which is how most soap was made 100+ years ago, will slow the saponification reaction. There are two reasons for this -- the soap will be easier to stir by hand and the saponification of the oils will be more complete.
If you add the lye too quickly, soap particles form quickly enough that they will make a solid mass at the top of the kettle. This soap mass is so thick and stiff that it cannot be easily mixed by hand. The water-salt-lye solution below the soap layer cannot be mixed with the the oils trapped in the soap layer, so the batch will not saponify efficiently. By adding the lye slowly, the soap mass remains more fluid so it can be stirred more easily, and you'll get better results with this type of method.
2. Is NaOH or KOH used nowadays? It looks like traditionally it was made by ashes, which would mean KOH. Would NaOH make a too hard soap?
Lye made from ashes is not necessarily entirely KOH. It depends on the vegetation -- what kind of plant and the location in which it grew. Certain plants that grow near salt water naturally have more sodium in the ash than other plants or plants grown inland.
Furthermore, the use of salt water (or seawater) in the soap making process provides an additional source of sodium. This free sodium reacts with KOH soap molecules to partially convert the KOH soap to an NaOH soap.
3. Is it really true that the soap is boiled at 200C (not F) as some sites are claiming?
That is absolutely incorrect. Boiled soap is a water-based system. Water at normal air pressure (0 bar gauge or 0 psi gauge) boils about 100 C (212 F). You'd have to have an expensive pressure vessel rated for well over 14 barG (200 psig) to raise the boiling point of a water solution to 200 C (392 F). Even a home pressure canner cannot get anywhere close to that. Bottom line -- the people who say that are confusing their Celcius with their Fahrenheith. That kind of pressure equipment ain't gonna be available with a handcrafted soap.
4. When fresh water is added, how do you exactly do it? Do you mix in the water properly? I mean, does not the soap solve up into the water and you end up with a gel? How do you prevent soap loss when draining the water? With how much water do you rinse it? Like same amount as soap?
The fresh water will indeed dissolve the soap. Dump it in and mix it up. This step is intended to reduce the amount of free lye in the soap (and also to remove the glycerin). This step has to be followed by a salt water rinse to drop the soap back out of solution. Yes, some soap is lost when you drain out the water.
The amount of salt required depends on the soap. Coconut soap will require more salt. An olive based soap will require less. How much water? How much salt? You will have to figure that out by trial and error. That is why boiled soap is as much an art as a science -- you have to have enough experience in this type of soap making to be able to "read" the soap and respond appropriately. I'd learn the technique by making a soap similar to the aleppo, but using cheaper materials.
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