I've never heard of any preservative effect of potash, the reason it is still used in traditional recipes is the ashy flavor it imparts. That smokiness is not to everyone's taste, baking soda is a perfectly good substitute if you prefer the end result.

You can refrigerate baking soda leavened cookie dough for 3-5 days without issue, it's batters where this would be a problem, but it would be whether you used potash or baking soda. The reason is that any chemical leavening agent starts to react with acid right away and produce carbon dioxide. Refrigeration slows this process, but doesn't stop it - cookies don't need as much leavening so it's not an issue whereas for a cake it's a problem.

Refrigerating a spiced dough for a couple of days is a good step because it lets the flavor of the spices come out and blend. You can achieve the same thing by mixing the spices with the butter and refrigerating the butter for a couple of days instead, the end result is the same from a flavor point of view.

This is a rather complicated issue, so I will answer it in steps.

Only oil and water

This is not something that is usually done in the kitchen, since the emulsion is not stable. But it is useful to consider this simplest version first before going on to the more complex ones. Let's assume that all you are mixing is water, oil, and maybe a bit of citric or ascorbic acid.

Here, the only influence of temperature is that you want the fat to be properly liquid when you start out. If you have fat that is solid (e.g. coconut oil) or cloudy (e.g. oilve oil you have kept in the fridge), let it warm up before you begin. In theory, higher temperature will be somewhat better, because the particle movement will give you a bit more mixing. In practice, you have to do so much forceful agitation to build the emulsion without any emulsifiers, that the contribution of higher temperature will be negligible. So the lowest possible temperature is given by freezing one of your components, and the highest possible is 100 C, where your water will evaporate.

You don't have any chemical reactions happening in this case. The pH doesn't matter.

So, in this case, the best temperature and pH are: any, within the typical range.

With an emulsifier

As above, it assumes that you are using water, oil, maybe some acid, and the emulsifier.

There is a wide range of emulsifers used in cooking, each of them with rather complex thermodynamic properties. Each of them has its "preferred" range of temperature and pH in which it performs best - in fact, some of them don't even work outside of a narrow pH range, and others require the presence of additional chemicals, such as sugar or calcium ions, to work at all. So nothing more specific can be said, it really depends on the exact emulsifier.

To find out the best temperature and pH range for a given emulsifier, consult a reference book on emulsifiers.

With protein

Traditionally, the only emulsifier frequently used in the kitchen was the lecithin contained in egg yolks. Also, many foods used in emulsions contain proteins by themselves. So you will most likely find yourself making an emulsion that contains proteins.

Here, it becomes more complicated. You have both emulsification and curdling going on at the same time. Your goal is to have the curdling happen, but at a slow pace, and not all the way, so the proteins will build a weak silky mesh interspersed with the oil bubbles of the emulsion, instead of having them create hard pieces separated by liquid.

In this case, temperature and pH become very important, and they are also interdependent. Also, it is not only temperature per se that matters, but (at least for eggs), the speed at which the temperature changes. Slower warming up leads to better sauce textures (that's why you should start with room temperature eggs).

If you do it without acid, you have to gently heat your mixture while emulsifying, and stop at the right point, before it curdles too much. For egg yolks, the ideal temperature is 72 Celsius, it will be different for other emulsifiers and/or other sources of protein. If you also add acid, it promotes curdling, and makes it difficult to achieve the right texture. So, texture-wise, the best pH will be neutral. But if you do have to change the pH out of safety or taste considerations, you will have to heat slower and to lower temperatures, and/or add buffers (e.g. sugar in a sweet emulsion) that counteract the acid effects.

This means that in the end, the pH will be determined by your recipe, and the optimal temperature will vary. The recipe is likely to give you vague instructions on how to heat (e.g. with a water bath), but not an exact temperature. So you have to go by the instructions and observe the process, using your experience to decide whether the heating has to be sped up or slowed down.