Hi, I am not working with dyes, so I can't provide answers to those specific aspects, but give you some general chemistry one.

1. Precipitation means that the dye is not interacting as well anymore with the solvent (water in your case) after adding the prec. agent. This is possible via 2 mechanisms. Either the agent interacts with the dye directly or it changes the properties of the solvent (most often via the pH value) and thereby weakens the interaction. From what I gathered on wikipedia, lake pigments most often work through the first mechanism. The prec. agents you add are salts, made up of positive charged cations and negatively charged anions. Normally, water is a great solvent for ions, meaning they are soluble. The dyes however are organic compounds with one part that is very ionic, meaning very water soluble, and one part which is not ionic at all, which in general means not water soluble.
   When you add the prec. agents, they first dissolve in water as well, but then its dissolved cations are interacting with the negatively charged groups on the dyes. That means the dyes are orientating themselves around the cations, which means that the non ionic parts are looking outside. Imagine it like a hedgehog rolling together -> only spikes on the outside and the spikes don't interact with water -> they precipitate.
   With that background: some dyes are better interacting with water than the prec. agent, which means they don't precipitate as well. For other dyes, the "outside" is still interacting well with water, which means again: no precipitation, because even the combined form can be dissolved.
   From what I gathered, the alum phosphate is not the precipitation agent itself, but rather to dilute the dye. But yes, your thinking is correct in that some groups better attach to the metal cation than others. Or since we are talking about organic compounds: The fundamental groups are the same (Hydroxy & Carboxy acid groups), but its rather the geometry of those groups that makes them latch onto the metal cation better.

2. Definitely! But it completely depends on the mechanism of the precipitation as explained in question 1. I saw calcium and strontium mentioned, so probably magnesium works as well.

3. Because you add something to the dye which you can't remove with cleaning: the precipitation agents! And you definitely don't want to eat aluminum salts! So no, as chemist: please don't eat anything where you added any salt besides table salt, if you don't know specific that it is allowed to be eaten (tofu is precipitated by sepcific salts and ok for consumption for example). also, it might be that the concentrated form of the dye is not healthy for consumption

4. Since I am not working with dyes, I can't answer you, but I was wondering what you mean here: Do you mean chemicals for precipitation or chemicals for dying?

5. Make your own fermented vegetables (sauerkraut, kimchi, raddish, ...)! You can season it exactly as you like and its delicious!

So sorry, a lot of "can't answer that", but I wanted to provide at least a bit of basic chemistry background. So let me give you an additional part of info you didn't ask for:

From what I found, one of the pigments in cranberries is cyanidin (wikipedia link) and that explains why you can't precipitate it: it doesn't have any carboxylic acid groups, which are normally the groups that most strongly interact with metal cations. But in the german wikipedia of the article it mentions, that the chloride version is nearly insoluble in water. So you could try adding simple table salt and see if that precipitates it. Second idea: it is also soluble in ethanol, so you could try and extract the berries with as high percentage alcohol as you have.