This is a pretty old thread. I will say that I add Sodium Lactate, Yogurt, and a good quantity of some form of sugar (for lather) to all my HP batches. Sodium Lactate and yogurt definitely do different things. Both help to make the soap batter more fluid. Yogurt is more obvious in that regard. My understanding, and I've only been doing this for a few months now, is that the humectant nature of sodium lactate helps to reduce water evaporation while cooking in addition to some other sciency stuff.
I also add some citric acid (and calculate the appropriate amount of NaOH to add) in order to add some sodium citrate (chelator) to my soap. Admittedly, I've not noticed much of a difference since the inclusion of sodium citrate.
It seems like the more "old school" way of doing HP is to put a crockpot on low and let it cook for 1-2 hours while disturbing it only occasionally to stir. I'm not sure why people don't want to stir except maybe they don't want it to cook too fast and cause a volcano? The hotter the batter is and the more you stir the faster the reaction happens. Personally, I don't like to use crockpots as it seems like the sides get really hot and you end up with more "crusties" than using a more gentle heating source like a double boiler which mostly heats from the bottom (I use a double boiler usually).
On the other end of the spectrum, if you cook your soap at 200-215F and stir constantly, you can cook your soap in ~10 minutes or less. Now, I stir a lot but do give it breaks where I cover with plastic and let it cook for a few minutes. If it takes me 15 minutes to cook, that's fine too. And in some cases, I've just used a microwave to heat up oils above 200F and completed the cook in under 10 minutes with fairly constant stirring with no additional heat needed. For that you will need a recipe engineered to react fast (think fast tracing). Note: Any of these "fast" ways of cooking will cause a rapid expansion (volcano) 100% of the time and you must be standing by to stir it down. It's best to have a container 2-3 times bigger than you need for this
I like to make it happen more quickly and I can usually pull out all my soaping stuff, make soap, get it in the mold, clean the kitchen, and put all my soapy stuff back up in under an hour. No right or wrong way to do it... just the way you like to do it.
All of my soap bars have been HP. Though I've been adding embeds to many of my recent bars and, for those, I make soap dough with CP.