Some thoughts here and sharing my experiences:
There is a secret to the "countertop hot process" that is not usually mentioned, and that is you need more heat than standard water content lye solution will give you. To get it to rip and roar like the demos, I had to reduce the water content to start with to make the lye hotter, keep the oils at 200F, and add some sugar to the lye water. Sugar is a common additive, but it's not usually mentioned in this context.
Using around 33% lye freshly made with a teaspoon or so of sugar PPO, plus HOT oil (200F, no less) will result in very fast saponification and a volcano or two. It really does get finished in ten or 15 minutes, every time. Full water or no sugar, it takes a lot longer and you have to keep the heat going.
Needless to say, it's stiff as all getout, the yogurt and some glycerin plus BOILING water after the cook will get you nice fluid soap. Don't get heavy handed with the glycerine, or you end up with sticky soap. 10 or 20 gr ppo is enough. Use sodium lactate, either in liquid after the cook or in the lye water as it does indeed help with keeping the soap fluid after the cook.
As far as fragrances go, there are two things to keep in mind -- the soap will be much lower pH and alkalinity than CP soap when you add the fragrance, and you get much less "morphing" of scents. That means a HP batch may not smell exactly the same as a CP one with the same FO. Only way to tell is to try it and see what happens. The other things is that you can reduce the amount of FO, possibly by as much as half depending on how it behaves. Again, you have to test to see what happens, it's impossible to predict exactly what will happen. Some will fade anyway, like citrus scents in general, but a good rule of thumb is significantly less FO is needed for HP. I have done a few batches of CP soap recently, and my high lard recipes gel pretty fast, and are zap-less as soon as they gel, so I'm expecting less morphing of scents and less loss. Generally takes less than an hour, but even than short time exposes the FO components to high caustic levels for a while.
Fluid HP will indeed take longer to dry as you are adding excess water after the cook. If you soap at 38% and can't keep from peeking, you will get very stiff soap and it will dry faster. No effect on actual cure, and I don't think that really varies between CP and HP, the actual required cure time is more a matter of the fatty acid profile than the process of making it. I have a soap made from used deep fryer oil (used for doughnuts only, not fish or chicken) and it's getting harder and whiter by the week. I suspect it will need a long cure, it's pretty high in oleic and linoleic acids.
HP soap tends to appear striated for me, it always has differential texture sort of like glycerine rivers. I suspect this is just variations in the initial water content in the mold, but it shows on cut surfaces. Cosmetic only, but noticeable.
Adding enough water to get fluid soap after the cook will result in more shrinkage, and this accentuates my rather erratic cuts. The soap sets up faster too, and will become quite resistant to cutting as soon as the center cools down enough to set. More water after the cook makes it softer longer, but I can usually cut mine within a couple hours of getting it into the mold. Don't wait past the next morning, it's gonna be quite hard by then, even with extra water.