Emergent Faction List

[Darth Venom]

I hope I can clarify some things from my experience here.

AFAIK there are two ways for a faction to emerge. There might be more, but I haven't seen any yet.
1. Rebellion in the relevant province, e.g. High unrest in French (or any other power than Westphalia itself) controlled Westphalia can lead to Hessia as a faction there.
2. Faction destroyed in its main theatre, eg. France is wiped out in Europe, Maratha in India, the US in America.... This will cause all provinces in other theatres go independent.

In both cases the faction owning the relevant provinces is not important; i.e. the emergence of the USA is not necessarily triggered by the destruction of either Britain of France. If Denmark happens to own all relevant provinces, nothing happens!
There are two types of emergent factions: Proper minor factions (USA, Hessia, Afghanistan, ...) and rebel factions. They don't appear in the diplomacy screen, have "XY Rebels" as name and can be treated like rebels in prevoius TW games (i.e. a kind of free-for-all landgrab )
Persia is a notable case because it is the only faction spanning two theatres without part of it being a colony. The game however does treat Afghanistan as a colony of Persia; it is the only one of its provinces in the Indian theatre (to verify go to diplomacy screen -> Persia -> provinces) and therefore becomes independant (rule 2. above) if the rest of the Persian provinces is conquered.

Hope this helps,

Darth Venom