Cauldron/bloodwrack

The last sentence is ambiguous in the case of Cauldrons/Shrines, as the Will of the Gods speical rule states (Dark Elves AB p. 47): "It can join units and leave as if it were a character," but then continues: "note that it is not otherwise treated as a character - it can just join and leave units like one."

However, I would argue that troop types do not change by something joining or leaving units. The troop type Unique unit is determined before the game starts, when building your Army list, by mixing different troop types through an upgrade (or something similar). They simply do not have the option ever to leave that unit - although all the models of one troop type can be destroyed, at which point the troop type will change from Unique to the troop type of the models that are left.

That would just postpone the question, until after the Hag is dead, while the Cauldron lives on. As such, it could leave and join units on its own.

Which is basically what I said...

Not quite. One can have both infantry and monstrous infantry models in one unit [or, more generally, of different troop types] without them changing to the troop type Unique. In which case, the models retain their troop types, and the Curse could be targeted at that unit, affecting the Infantry models, but not the MI models.

Probably not, as giant rats and their packmasters are the same size and effected the same by most things.

Infantry and monstrous infantry are different sizes, and are affected by different rules differently. Thus, making them troop type 'unique' prevents a lot of arguments about 'does this rule work like a, or like b, because we have both a and b in the unit?'

I should emphasise, though, that this matters only for effects that specifically target units of a certain troop type, not models. So, the Packmaster/Giant Rat mixed unit will not be affected by the Curse, or stomp, but the Giant Rats as War Beasts will e.g suffer fear from Flaming Attacks.